text:
Whitestone REIT (WSR) Q3 2020 Earnings Call Transcript
By Motley Fool Transcribers – Oct 27, 2020 at 1:30PM
NYSE: WSR
WSR earnings call for the period ending September 30, 2020.
Image source: The Motley Fool.
Whitestone REIT (WSR 0.59%)
Q3 2020 Earnings Call
Oct 27, 2020, 8:30 a.m. ET
Prepared Remarks
Call Participants
Prepared Remarks:
Good day and welcome to the Whitestone REIT's Third Quarter 2020 Earnings Conference call. [Operator Instructions]
And at this time, I would like to turn this conference over to Kevin Reed, Director of Investor Relations. Please go ahead.
Kevin Reed -- Director of Investor Relations
Thank you, operator. Good morning and thank you for joining Whitestone REIT's third quarter 2020 earnings conference call. Joining me on today's call are Jim Mastandrea, our Chairman and Chief Executive Officer, and Dave Holeman, our Chief Financial Officer.
Please note that some statements made during this call are not historical and may be deemed forward-looking statements. Actual results may differ materially from those forward-looking statements due to a number of risks, uncertainties and other factors. Please refer to the Company's earnings press release and filings with the SEC, including Whitestone's most recent Form 10-Q and Form 10-K for detailed discussion of these factors.
Acknowledging the fact that this call may be webcast for a period of time, it is also important to note that this call includes time-sensitive information that may be accurate only as of today's date, October 27, 2020. The company undertakes no obligation to update this information. Whitestone's third quarter earnings press release and supplemental operating and financial data package have been filed with the SEC and are available on our website www.whitestonereit.com in the Investor Relations section.
During this presentation, we may reference certain non-GAAP financial measures, which we believe allow investors to better understand the financial position and performance of the Company. Included in the earnings press release and supplemental data package are the reconciliations of non-GAAP measures to GAAP financial measures.
With that, let me pass the call to Jim Mastandrea.
James C. Mastandrea -- Chairman and Chief Executive Officer
Thank you, Kevin, and I want to thank you for joining us on the third quarter investor call. We continue to hope and pray that all of you, your families and your businesses remain healthy, and are doing well as we navigate these most unusual times. I am pleased to share with you our strong operating and financial results that have sustained the economic downturn caused by the COVID-19 pandemic. Our business has recovered quickly, and in doing so produced shopping center sector-leading results.
Dave Holeman, our Chief Financial Officer, will provide a more detailed look at the drivers of our operating and financial performance. We attribute these results to owning properties located in areas with high household income neighborhoods and the fastest-growing MSAs in business-friendly states. Operating a consumer business-driven model based on consumer demographics and psychographics to design a tenant makes that drives customer visits and experiences.
Our strategic customer focus differentiates Whitestone from other real estate owners that leads to traditional hard and soft good retailers. To enhancing our intrinsic value through redeveloping and developing, which adds value to -- value as we physically expand our real estate footprint and grow rents and to balancing a capital management structure to drive revenues, net operating income and funds from operation.
With this background, I would like to provide some color as to where we are today, where we plan to go in the coming months and quarters to drive long-term shareholder values, key building blocks to Whitestone's success and our progress toward the long-term goals we announced in February of 2018. First, where we are today. Since the onset of COVID-19, which injected considerable economic uncertainty, relative to the real estate industry in particular, we've had shopping center sector leading rental collections throughout the pandemic and most recently announcing 90% cash collections for the third quarter of 2020. These results are in line with our expectations and are trending upward from the first-half of the year.
Rental collections, solid leasing spreads and execution of our business model is deeply rooted in Whitestone's team culture. Since the end of the first quarter, we focused on improving our dividend payout ratio, increasing our cash on hand, paying down our debt and strengthening our balance sheet. These actions have prepared us for the next level of growth and provided greater financial flexibility.
Our judicious and disciplined allocation of capital has produced positive third quarter operating results, including occupancy of 88.9% at the end in the quarter, revenue that approached $30 million for the quarter, up 8% from the second quarter. Property net operating income of $21.3 million, up 6% from the second quarter. Leasing spreads that were strong and resulted in a positive 11% increase for the quarter.
Cash collections, which I mentioned earlier that achieved 90% for the quarter were up from 81% in the second quarter and funds from operation core that were solid at $0.23 per share, up $0.01 or 5% from the second quarter. Dave in his remarks will follow mine, will go into greater detail on these results and comparisons to the prior year.
Overall, I am very pleased with the way our management team is navigating the economic crisis caused by the coronavirus, stabilizing our business quickly and enabling us to refocus on extracting the embedded intrinsic value in our properties, while targeting acquisitions that are accretive and further scaling our platform.
Second, where we plan on doing in the coming months. We intend to maintain our disciplined capital allocation philosophy going forward, as we have over the past. Our track record of our growth from 2010 to 2019 speaks to our plan. Our long-term plan is to grow our strategically chosen markets of Houston, Dallas-Fort Worth, Austin, San Antonio and Phoenix through redevelopment and development opportunities within our portfolio and externally grow by making targeted acquisitions in our existing markets and beyond.
Third, the key building blocks for Whitestone's success is building a portfolio of choice properties in great markets that we lease and manage with a team of well-trained professionals who continue to do what has worked for us in the past. We view Whitestone as a highly differentiated and somewhat a special really -- specializing in e-commerce-resistant entrepreneurial tenants and crafting the right mix of those tenants who ultimately meet the neighborhood consumers' needs. Our diversified mix of tenants provide essential services for lease that cannot usually be acquired online, if at all. As a result, our open-air centers provide community experiences that cater to adjacent neighborhood as an extension of their lifestyle.
While we recognize that we are in the retail real estate business, our differentiated approach and contrarian business model has allowed us to profitably grow an asset base over the years despite significant overall industry challenges. Our growth since our IPO in 2010 has been steady and disciplined and we intend to continue this disciplined approach.
Finally, we continue to make progress toward our long-term goals of improving debt leverage and scaling our G&A cost that we set in 2018. During the third quarter, we paid off $9.5 million of real estate debt from cash, lowering our net real estate debt by $11 million from a year ago and improving our ratio of debt to gross book value real estate assets to 55% from 58% a year ago. We also are making progress on scaling our G&A. These costs since the beginning of the year have been reduced through reduction of headcount from 105 to 85 or 21% through further automation, improvements in processes and all of our team working smarter and harder. Our total G&A costs for 2020 are approximately $900,000 or 6% lower in the same time last year.
With that background, I would now like to turn the call over to Dave Holeman, our Chief Financial Officer. Dave?
David K. Holeman -- Chief Financial Officer
Thanks, Jim. First, I would like to take this opportunity to thank all of our associates to continue to produce the best results possible, given very difficult times. We have a highly dedicated team that works every day to create local connections in communities that drive and we feel strongly that we are positioned to withstand the current headwinds and thrive into the future.
Given the severe economic pressures caused by the coronavirus during the quarter, our portfolio has performed quite well. Despite having a significant amount of our tenant businesses impacted, we only had a handful of tenants close for good such that the portfolio occupancy rate held up well ending the quarter at 88.9%, down just 0.3% or 13,000 leased square feet from the second quarter. Also our annualized base rent per square foot at the end of the quarter was $19.43 and $19.38 by cash and straight-line basis. This represents a 0.5% increase on a cash basis and a 1.3% decrease on a straight-line basis from a year ago. The decrease on a straight-line basis is largely related to the conversion of 84 tenants to cash basis accounting and the associated write-off of accrued straight-line rents.
Our square foot leasing activity was up 43% from the second quarter of 2020 and 46% from the third quarter of 2019. And we are pleased with positive blended leasing spreads on new and renewal leases of 3.3% and 11% on a cash and GAAP basis for the quarter. As Jim mentioned, for the quarter, we collected 90% of our rents, this include base rent and triple net charges billed monthly.
We have also entered into rent deferral agreement on 3% of our third quarter rents. As part of the deferral agreements, we have negotiated beneficial items, such as, entry into our online payment portal, further reporting of tenant sales, suspension of co-tenancy requirements, loosening of exclusive or restrictions that allow further development and stronger guarantees.
While we are encouraged by how things are progressing, the pandemic has continued to impact our business in term -- in the terms of our financial results. Funds from operations core for the third quarter was $10.1 million or $0.23 per share compared to $10.9 million or $0.26 per share for the same quarter of the prior year and our same-store net operating income for the quarter decreased by 4.5%. These decreases are primarily due to the impact of the pandemic, which resulted in a charge of $1.3 million to bad debt expense and $100,000 in write-off of straight-line receivables in the third quarter. This charge was incremental to the bad debt expense and straight-line revenue recorded in the prior year by $900,000 or $0.02 per share.
Let me add some color on our collectability analysis related to the pandemic and the related receivable balances. At the end of the quarter, we had $23.6 million in accrued rents and accounts receivable. This consists of $21.1 million of billed receivables, $1.9 million of deferred receivables, $16.1 million of accrued rents and other receivables and a bad debt reserve of $15.5 million. Since the beginning of the year, our billed receivable balance has increased $4.4 million and our deferred receivables have increased $1.9 million for a total build and deferred receivable balance increase of $6.3 million. Against this increase of $6.3 million, we have recorded an uncollectible reserve in 2020 of $4.5 million or 71%.
Our accrued rents and other receivables had decreased by $1.1 million, largely the result of the write-off of accrued straight-line rent receivables on tenants converted to cash basis accounting and our bad debt reserve has increased by $4.3 million.
In accordance with generally accepted accounting principles, if the company determines that the collection of a tenant's future lease payments is not probable, the company must change the revenue recognition for that tenant to cash basis from accrual basis. In light of the financial pressures that COVID-19 has been placing on many of our tenants, we reevaluated all of those tenants in the second and third quarters, as a result have switched 84 tenants in our portfolio to cash basis accounting. These 84 tenants represent 3.6% of our annualized base rent and 3.4% of our leasable square footage.
As a result of this conversion to cash basis accounting we have written off $1.1 million of accrued straight-line rents for the year, but the company intend to collect all unpaid rents from its tenants to the extent possible. Our tenants on cash basis accounting paid 63% of contractual rents in the third quarter, up from 41% in the second quarter. We have provided some additional details of our collections, it can be found on Page 27 of the supplemental.
Turning to our balance sheet. Since March, we have implemented various measures to conserve cash, including further reductions in headcount. To-date, we have approximately $39 million in cash, representing a 6% increase since March 31st. Additionally, we paid off $9.5 million of real estate debt in the third quarter and have no debt maturities in 2021. We have reduced our total net real estate debt by $11 million since the third quarter of 2019. Currently, we have $111 million of undrawn capacity and $13 million of borrowing availability under our credit facility. We are in full compliance with our debt covenants and expect to remain so in the future.
Looking ahead, one particular trend we are seeing as COVID-19 persists is that Whitestone's best-in-class geography is benefiting from net migration of businesses coming out of the regulation-heavy gateway market. Our markets continue to attract both large and small businesses, as evidenced by Charles Schwab's recent announcement of its move of its headquarters to Dallas from San Francisco.
Whitestone's properties are seeing this migration also. Recently, we added a successful restaurant who needed to escape the high-tax high-regulation environment of California to our Mercado property in Arizona. Examples like these prove the resilience of our markets and give us further assurance that our business model is thriving. We have seen pent-up demand in our markets, as consumers are leaving their homes and returning quickly and in force. Parking lots are filling and stores and restaurants are becoming more active. Whitestone is well positioned to capture this pent-up demand and intends to do so.
Our team has worked together through this ongoing crisis and our shareholders will reap significant future benefits through greater collaboration, a more robust exchange of ideas, better and more effective communication and improved systems and processes that provide new actionable data and allow us to more efficiently scale our infrastructure. Whitestone is continuing to perform and deliver on our strategic plan. We operate in many of the most highly desirable growth markets in high-population growth states and expect these markets to lead the country in economic recovery from the pandemic. We look forward to providing further updates as we progress.
And with that, we will now take questions. Operator, please open the lines.
[Operator Instructions] And we will take our first from Michael Diana with Maxim Group. Please go ahead.
Michael Diana -- Maxim Group -- Analyst
All right. Jim, Dave, can you hear me?
Hey, Michael. Can you hear me?
Yeah. Okay...
Yeah. Thank you.
So I'd like to ask first about leasing, which looked very strong. I noticed your new leases, you did 32 new leases. Where are those coming from? I mean, I know your general strategy. Could you be a little more -- give us a little more detail on what sort of tenants you're tracking to your new leases?
Sure. Yeah.
Hey, Michael, this is Dave. Thanks for your question. Yeah, we are very pleased with the rebound in our leasing activity. As you mentioned, we've seen -- from an activity perspective, we've seen the third quarter coming back to kind of the first quarter in pre-pandemic level. We've had good activity and then good spreads in there, as well.
In our markets, I think we've seen resilience pretty equally among our markets. I would say the Dallas market is doing very well. And then from a tenant-perspective, we continue to see demand from kind of experiential service space tenants. We've seen a lot of activity on the pet level, we've seen restaurants continue to -- good operators continue to look for opportunities. So, I think, we're seeing demand across our unique and differentiated tenant mix and I think in our markets, they all appear to be recovering it very nicely.
Right. And the one restaurant you mentioned from California where -- was that in Arizona or in Texas that they came to?
Yeah, that's the one I mentioned in my -- this remarks was actually to Arizona, obviously, with the proximity of our Phoenix markets, California, we've seen significant -- seen migration over the years and continue to see that. And then we're seeing a fair amount into Texas as well from some of the coastal cities where people are just looking for places where they can run their businesses a little easier.
Right. Okay. And then the second area I want to talk about is your bad debt/uncollectable revenue, you -- as you said, you reviewed all your tenants, I mean, very bottoms-up. What's on the tenants that you talk to when you go through that process? What is their outlook? I mean, are they pretty optimistic about what's going on? Or how long do they think things are going to remain challenging?
I'll start and Jim will probably give some overall comments as well. I think it's obviously a fluid and granular process right now. I think, we are seeing a lot of pent-up demand in our markets and people kind of coming back in toward the restaurants and other areas, schools are open in all of our markets. Most of those school districts are offering kind of in-person school as well as some virtual, but we're seeing demand and I think with that, we're seeing some optimism from our tenants. We're seeing tenants in our markets that are a little bit surprised by their activity levels. We have a lot of tenants that have been creative through this pandemic, restaurant operators that have created new forms of revenue. We've had restaurants that historically have not done curbside or take-outs have developed that and I think they'll continue to have that as -- in their portfolio, going forward.
Yeah, and I'll add to that, Michael. We're seeing some second-generation restaurant space, both in Arizona and Texas that is leasing up very quickly. In other words, with all the equipment there and all the tables and things and we've had been fortunate being able to -- if people don't pay, we lock them out, but the restaurants are ready to reopen again very quickly. So we're seeing that happen in that space real quickly.
Second thing we're seeing is that we have a product called CUBExec, which is small offices, enclosed offices, and we have sections of these in our different retail properties and what we're finding is, those are leasing up. It might be 150 to 200 square-feet, see that. We also did a new lease with a Pilates studio and we took a 25% equity position in that and it's a person who teaches teachers. So with that we have -- we will have franchise rights to open that in multiple properties and what we're seeing is a strong and maybe even a resurge in demand for the exercise called Pilates, particularly on the -- in the Texas and Arizona areas.
Okay. So given you -- that there is pent-up demand and there is some optimism here, what -- your bad debt reserve cost you $0.03 a share this quarter, do you see that fading away fairly quickly? Or...
Yeah, I think we are seeing improvement. I mean, our bad debt on collections for the third quarter were 90%, up from 81% in the second quarter. Obviously, there is predicting the ultimate -- when we get to the end of it is very difficult, but I think we are very pleased in the positive trends we're making. We don't want to call the end to it, but we are pleased with the progress we're making. We also reported our October collections, which were 90% through the 25th of the month. So I think that bodes well. So that being above our set -- our third quarter amount of well. So, I think, we expect overtime our collection percent to go up. With that, the related reserve rent collectability to go down.
Okay, great. That's all I have. Sounds good. Thank you.
Thanks, Michael.
And our next question will come from Craig Kucera with Riley Securities. Please go ahead.
Craig Kucera -- B. Riley FBR Inc. -- Analyst
Thank you, good morning, guys. Can you give me some additional color on what specifically is within the entertainment bucket on table -- on Page 27 in the table? Is that mostly movie theaters or any other categories there?
Yeah, hey, Craig. So entertainment makes up about 2% of our leased square footage, about 2% of our annualized base rent. The tenants in there, we've got, I think, we got one movie theater, we've got some -- we've got one large, I can think of child kind of entertainment venue, we've got some local theaters. So probably movie theater is the largest one and then we have community theaters and we have a like a children play area kind of entertainment facility.
Got it. And I just want to double check on the expansion of your line of credit capacity this -- from last quarter, I think it increased about $12 million. Was that due to some improvement in NOI sequentially or was that because you paid down the $9 million mortgage or something else?
Yeah, so our credit facility availability did improve this quarter by about $12 million. That's a result of the two things you mentioned, one is our NOI is a little better, obviously with greater collections. And then we did pay off one mortgage loan, which allowed that property to be unencumbered and therefore included in the availability of our property. So that represented -- that mortgage loan was about $9 million, the creditability went up about $12 million, so about $9 million of it was from paying off debt and the rest was from improvement in NOI.
Okay, got it. Jim, I feel like early on, you worked hard in assisting your tenants in garnering government support and I think in late first quarter you said that about 40% of your tenant base had reached out and do you have a sense of how many of those tenants actually were successful in getting into programs such as the PPP as you were working with them.
Yeah, in terms of the PPP, we've had very, very good success. I'd say upwards of 80% at least and it really paid off because they -- these are small business people, and when I say small just in square footage size, usually their net worth is in excess of $1 million to $5 million, but they just -- they are not attuned to how do they work with these different banks. We were able to set up banks for them to really going and expedite that process and it's really what got really well for them.
Got it. And so what was -- somewhat unique with your collections process over the past couple of quarters is that you've got new collections. I think in second quarter it was 81%, you deferred 5%, this past quarter you got 90% and you deferred 3%. I guess, with the bucket for example this quarter of a 7%, kind of, are you thinking those are longer term tenants or are those just based on the fact that you've not -- it doesn't appear abating anything. Can you kind of give us some thoughts on the folks that aren't paying sort of their longer-term liability.
Yeah. So, I think, as I said earlier, it is a very fluid process. I think our business model has shown very well and one of the things about our business model is we have probably a little more hands on debt. We're not spread out like some of the others where we focus on markets. In those markets, we focus on specific tenants. So we have a lot of depth and relationships in the markets we operate. We've worked with those tenants. Many of those have got their businesses go on, the revenues go on and they paid us. Some of those, we've reached agreements where we allowed them some relief for a period of time and they paid us back -- will pay us back over 12 to 24 months. And then some frankly at this point are just -- we're working with. So I don't think that in the quarter, we had 90% collections, we had 3% deferred, at least kind of 7% that we're working with and continuing to have them -- try to help them operate their business and pay us.
So I think, we're pleased. If you look at our accounts receivable, I think, I reported some of the stats in our measures, we're seeing good progress in collecting and not letting that accounts receivable balance balloon up. We've been very diligent in collecting and also in doing the rent relief agreements, we've also got some favorable things that will benefit us in the future with greater sales reporting of tenants, greater entry in our online payment portal and we've also cleaned up some of the leases where they were exclusive that we would like to have a little better.
Yeah. I know it...
I'll add to that. Overall our properties are really very, very good and there are -- a lot of them are replicated from city-to-city and the process are replicated as well. So when we have a tenant that really looks like they're not going to make it, we just move very quickly to get them out of the space and we're able to collect anywhere from 75% to 90% of what they might owe us on the contract period of rent and then we replace them with someone else. So that's what got very well.
We've also found that we've opened our doors to some venues that have helped drive traffic in certain properties. For example, there is a -- in Arizona, there is a group called Coffee & Cars, which is a number of exotic high-end cars and these folks like to get together usually on the first Saturday of a month from 7 a.m. until 10 a.m. and so we've been able to capture that from some other properties within the area that have not let that continue during COVID. So we were able to open during COVID, now we capture that and we get anywhere from 40 to 100 cars there on the first Saturday of every month and they're probably range there from about $25 million to $30 million worth of cars.
So that's really stimulated our restaurants in certain properties where they open early in the morning, they serve tacos and breakfast and then folks who used to come back for night [Phonetic], so we're doing some thing that we continue that process of how we can serve the tenants.
Okay. Thanks, that's it for me.
Thanks, Craig.
[Operator Instructions] And our next question will come from Aaron Hecht with JMP Securities.
Aaron Hecht -- JMP Securities LLC -- Analyst
Hey, guys. You talked about the strength of your Arizona and Texas markets and communities, wondering if you had any metrics that you could comp the performance of those properties versus pre-COVID levels, whether it be foot traffic or revenue, anything to give us perspective there?
Yeah, I think that's a good question, Aaron. I think a couple of metrics, one is just the leasing activity obviously is a metric we follow to see the demand from ultimately consumers driven -- consumers then drive demand of the tenants. So we've seen our leasing activity back to kind of pre-COVID levels in the quarter. So I think that's very positive.
We do a lot of work with looking at traffic levels in our centers and I don't have those numbers here, but I'll tell you anecdotally we are seeing in our markets, the traffic counts in our center, the consumer visits coming back to kind of pre-COVID levels. We use different software to do that, but I don't have the numbers, but anecdotally we are seeing that level.
As you know, we open all -- we operate all open-air centers and so we're also in markets that are tend to be warmer geographies. So we're very pleased with the activity and we look forward to actually the next few months, where our markets are great times or people are able to enjoy themselves in our centers, sit outside. Our tenant mix is really positioned for the long-term with not a lot of big boxes, more experiential tenants. And so with our real estate that we think is really in best-in-class geographies and our tenant mix, we feel very good about the activity level we're seeing.
On the tenants, so you're continuing...
I want to add.
Sorry, Jim. Yeah.
Well, I was going to add to that. It's interesting because our experience with COVID that was started back in March, we have these four offices, one in Houston, Phoenix and Dallas and San Antonio, we had mentioned earlier roughly around 100 people. We've only had one person come down with COVID in those four locations and two states and that person has made a trip back to Seattle during some of the events or going back, we all watched the events on TV and came back and she had COVID and then came back to our Phoenix office and then resign -- before she -- as you saw she never really came back in office, but only one person of all those offices. We thought that was kind of interesting.
So our experience was very, very slim in that. The other thing we noticed is the influx from people from California, you had the COVID and the shutdowns, which were a little bit more dramatic than in Arizona and following that were the fires. And so when COVID was just reopen, there was lot of fires in California in [Technical Issues].
And the smoke was intolerable in most places like San Francisco in places like that is as you probably know Aaron being out there. And so we had a lot of people found to come over, some are going back, some are not. So we're starting to see some activity, I think we have had like eight interested folks leaving who had moved here from California.
On the collection side and I guess it does to ultimately leasing side as well. Obviously, 90% very strong, stable through October. Are there -- Is there structural impediments, given the tenant base or the size of space that you have at your communities that will make that last 10% being significantly harder to sort of collect any time soon without a fair care because maybe the movie theaters, something like that kind of any insights there into any structural difficulty that may be more difficult to get you to that file 95% to 100% as they're getting?
Yeah. Great question. Yeah, we've made -- we're pleased with the progress we're making. As you said, we've 90% in the third quarter, up from 81%. Obviously, now the goal is to get that back to normal, up in the high 90s. There really aren't any structural impediments. I think we're very well positioned with the tenant base that's largely smaller space tenants, entrepreneurial, we don't have any one tenant that makes up more than 3% of our revenue.
So I think it's just going to be continued hard work. I think the next few months, as I mentioned, in warmer climate markets like we operate in will be our great months for us. So there are no structural -- there's is no big lease that's between that 90% and 100%. It's just like our normal tenant base, which is smaller tenants, service-providing tenants, entrepreneurial tenants.
In those small tenants, are entrepreneurial type tenants in most cases where their business is important. You'll see them there 12 hours a day, not punching a clock from nine to five. But just to remind our listeners that in Texas and Arizona, we have business -- there are favorable state laws that permit us to lock a tenant out in seven days if they haven't paid their rent and that's something that tenants know that we can exercise and their business is very important to them. In other states, particularly up in the mid-west and on the West Coast and in New York, it's usually a 90-day period to lock them out and you have to go to court and you have to hire a lawyer. So the rights are very different in Texas and Arizona that's why, when we look beyond that for expanding, we look at Tennessee, for example, and Florida and Colorado. They have business-friendly laws as well. So I think that helps a lot because the tenants know it's there and we know that it's there and we're very judicious about how we use that.
And then on the Pillarstone portfolio, looks like occupancy drops. Is there any comment around that we -- how we should think about performance?
Yeah. The Whitestone still has an ownership interest in eight properties that are kind of non-retail properties that represents, I think it's about -- it's a small amount for us, I think for the quarter it was $200 million impact to earnings. So I think that portfolio ultimately, Pillarstone has some plans to develop and change the use of those assets. So we are seeing the occupancy trend down just a little bit. It's not a large impact to Whitestone at this point, it's about a couple of hundred thousand of impact to FFO this quarter.
Thanks, Aaron.
[Operator Instructions] And with no further questions, I'd like to turn the call back to Mr. Mastandrea for any closing remarks.
Well, thank you, operator, and thank you all for joining us. I would like to close by just reiterating that what we find is that the proof is in the pudding. Our management team and our Board are significant shareholders and we are shoulder-to-shoulder with our shareholders, all of you, as we strive to make our existing and our new shareholders aware of our story. We have a great story and we love sharing it. When we're able to accomplish this, we expect to break away stigma that really has dragged down the real estate retail industry.
We believe that our operating strength is evidenced by our track record will continue to improve as the economy gains traction and guide our Board's decision to adjust our dividend payout ratio and dividend payout and while in the meantime, we will judiciously continue to pay attractive dividends as we have and in the future.
As shareholders, you're all aware that our share price does not reflect our business operations that have been dramatically impacted by the COVID pandemic. The superiority of our business model and the strength of our management team's experience has really stood the test and I'm really pleased with that. During the past six months, it has and it continues to produce exceptional financial results. Our responsibility is to continue to perform and make the market aware of our story. We think that's the linkage that we have to really work on as a team here.
We believe that as the market becomes more knowledgeable of what we're doing, they will see Whitestone as an investment of choice. When these dynamics intersect, we expect our share price to more accurately reflect our true value. We thank our shareholders for the confidence they have placed in us and our stewardship and as we move toward the future, we remain excited and we're really -- the team are very optimistic of what we see ahead and to serve you by providing the results we have and continuing to work to provide industry-leading returns.
We will still recognize there will be some macro-driven volatility in the near-term, but our deep experience and strong performance during the pandemic are reminders that there is a bright, long-term future in front of us. We don't, in any of our numbers or in any of our expectations included tailwind in our plan, but if we catch one, it will really amplify tremendously the results that we've experienced and shared with you.
As we move ahead, we will continue to hold ourselves accountable and we intend to stay true to our core values and to execute on our strategic business model and we intend to deliver to our shareholders significant, long-term sustainable values. We know that God's hand is on our shoulders as we focus on serving our shareholders, employees and our stakeholders and we thank you all for your confidence in us.
With that operator, we will close.
[Operator Closing Remarks]
Call participants:
More WSR analysis
All earnings call transcripts
This article is a transcript of this conference call produced for The Motley Fool. While we strive for our Foolish Best, there may be errors, omissions, or inaccuracies in this transcript. As with all our articles, The Motley Fool does not assume any responsibility for your use of this content, and we strongly encourage you to do your own research, including listening to the call yourself and reading the company's SEC filings. Please see our Terms and Conditions for additional details, including our Obligatory Capitalized Disclaimers of Liability.
Motley Fool Transcribers has no position in any of the stocks mentioned. The Motley Fool has no position in any of the stocks mentioned. The Motley Fool has a disclosure policy.
Whitestone REIT Is Rallying. Should You Jump In, Too?
Avoid This Stock Even If the Market Crashes


text:
As a member of the Methodist Co-operative Society Ltd, you are entitled to numerous privileges and benefits which were launched using the Common Good Fund (CGF), which was set up to allocate a portion of our yearly surplus for the benefit of members of the society.
As set out under clause 7.6 of the Methodist Co-operative Society Ltd's By-Laws, the CGF shall be used for the benefit of members of the Society and may be devoted to any educational, medical, social or charitable purposes. The CGF is derived from annual appropriation of the net surplus of the Methodist Co-operative Society Ltd.
Some of the benefits are listed below:
To cheer members, we send a fruit basket to any member who is hospitalized in Singapore, provided the MCSL office is informed of the hospitalisation.
The CGF Committee organizes for members a wide range of subsidized events, educational talks and outdoor tours. Past organized events include the following: D-I-Y Massage, Chinese Tea Appreciation, Christmas Party, Beauty & Health Workshop, 'Healthy Eyes for Life' Eye Care Forum, Vegetable and Fish Farm Tours and Food Processing Factory Tours.
Membership Sub-Committee
A sub-committee on membership was established with clearer focus in July 2008 to serve our members better for the years to come. Currently, we have tie-ups with other co-operatives and organizations to give special rebates for services providied by them. Click here for more. We welcome suggestions from members on other ways in which the co-operative can better serve its members. More beneficial schemes may be explored and launched for members.
You are entitled to receive annual dividends from the shares you own. Over the years, we have consistently declared and issued around 4% dividends. It is a token of our appreciation for your love and support through the years.
Bible Daily
Proverbs 10:28 - "The hope of the righteous shall be gladness: but the expectation of the wicked shall perish."
496 Upper Bukit Timah Rd #04-21
John Wesley Centre
Tel: 6767 1635 / 1636
Buses to John Wesley Centre
Along Hillview Road -
Along Upper Bukit Timah Road -
67, 170, 171, 178, 184, 961
Parking of vehicle in the centre is free.
Copyright © 2023. Methodist Co-operative Society Ltd.
Careers @ MCSL

redpajama_set_name: RedPajamaCommonCrawl


Fox News Back At It, Brand Radiohead the "Poor Man's Coldplay"
More earth-shattering insight from music's most finger-on-the-pulse minds.
The band's one-off song is dedicated to those affected by Hurricane Harvey.
Turns out that "sad and brooding" is still a good look on these guys.
"I might as well go home now": Watch this 19-year-old German kid dazzle Chris Martin.


text:
Simon Hewitt Cook (born 29 January 1972) is an Australian former cricketer.

After graduating from the Australian Cricket Academy, he debuted for Victoria early in 1993. He showed promise in the 1993–94 season, taking 30 wickets at 29.5.

He moved to New South Wales in the 1995–96 season. He was a surprise selection in two Test matches against New Zealand late in 1997, despite being considered by some to be not even up to first-class standard. On his debut, he took five wickets in the second innings against New Zealand in Perth.

Soon after his two Tests, which yielded a respectable bowling average of 20 as a replacement for Glenn McGrath, Cook disappeared off the scene fairly quickly due to injuries. He did play four more Sheffield Shield games in that 1997–98 season, including two four-wicket hauls, but in 1998–99 played just two games for just one wicket. He missed the entire 1999–2000 first-class season, and his next match in December 2000 turned out to be his last in the Pura Cup/Sheffield Shield.

Cook retired from all forms of cricket at the end of the 2003–04 season, playing with Victorian Premier Cricket's Melbourne Cricket Club.

References

External links

1972 births
Living people
Australia Test cricketers
New South Wales cricketers
Victoria cricketers
Melbourne Cricket Club cricketers
Worcestershire Cricket Board cricketers
Cricketers who have taken five wickets on Test debut
Australian cricketers
Cricketers from Victoria (Australia)

redpajama_set_name: RedPajamaWikipedia


Belanger Books is teaming with The Junior Sherlockian Society to bring you the first anthology of Sherlock Holmes short stories for student readers. The anthology is titled The Irregular Adventures of Sherlock Holmes and will feature stories by student writers as well as professional authors. Net proceeds will go to the Beacon Society to help them expose more students to the original Sherlock Holmes stories written by Sir Arthur Conan Doyle as well as newer stories about the great detective.
Requirements: All entries must be at least 1,000 words in length, follow Derrick Belanger's 10 Rules for Writing a Sherlock Holmes pastiche (http://juniorsherlockian.com/wp-content/uploads/2017/08/10-Rules-for-Writing-a-Sherlockian-Pastiche.pdf), and be rated "PG" (i.e. suitable for a school library).
Authors retain the rights to their work.
Payment: Authors shall receive a paperback and electronic copy of the anthology for letting us use their work in The Irregular Adventures.
Due Date: All entries must be received by February 15th, 2019.
Please send entries to [email protected] or to [email protected]. Make sure to include your name (first and last), title of your story, story length, and attach your story as a word document. If you are a student also include your grade and name of your school. Writers included in the anthology will be announced in March. All authors will receive a complimentary copy of the anthology and will retain full rights to their stories. If you have questions please contact us at [email protected].


An Oregon driver is being sued for $3.5 million after he ran over a woman in a grocery store parking lot. According to the court documents, the man was on the "Whole 30" diet plan and was lightheaded when the accident occurred.
On September 21, 2017, Susan Matthies shopped at Safeway grocery store. After she finished her shopping trip, she went out to the parking lot to load the groceries into her car. While she packed her car, Robert Morgan pulled into the parking lot.
According to the lawsuit, Morgan was following a strict diet plan called the "Whole 30." The suit alleges that Morgan skipped breakfast and lunch on that day. He missed the meals because he was returning home from a trip to Amity, Oregon. At the time of the crash, Morgan was lightheaded and nauseous.
As he arrived at Safeway, he pulled into the parking lot. He failed to see Matthies and he drove into her car. After he hit Matthies' car, he proceeded to run her over.
After the accident, an ambulance transported the woman to the hospital. She passed away that evening from the injuries she sustained from the car.
Following Matthies' death, police investigated the accident. Evidence police collected at the scene and through interviews determined that Morgan was not intoxicated nor criminally negligent.
Unsatisfied with the result of the criminal investigation, the family of Matthies decided to file the civil lawsuit against Morgan. Matthies' estate is suing Morgan for negligence associated with her death.
Matthies' estate is asking for damages to cover funeral and medical expenses. Additionally, the estate says by losing their loved one they want non-economic damages of $3.0 million. In total, the suit is requesting $3.5 million from Morgan for the unfortunate accident.
While the family has a right to be upset about Matthies death, I'm not sure that the "Whole 30" diet is to blame. Morgan's hunger wasn't a result of the diet but rather that he skipped two meals that day. Instead of eating on his trip home, he attempted to hold off until he got to a grocery store.
Perhaps the estate is attempting to blame the diet because Morgan would have had a tough time finding food to meet the program's guidelines. The "Whole 30" diet eliminates many ingredients, drinks, sugar, and grains. Fast food would not have been an option for him on his trip home.
Even with that excuse, the lawsuit seems like a bit of a stretch. Should he have eaten before he left that day? Probably. Was he negligent for continuing to drive while hungry? I don't know.
With the criminal case already closed, the civil lawsuit is the final attempt of the family to seek justice for their loss. Civil suits have a lower burden of proof than a criminal trial. Therefore, a judge or jury may side with the family.
We will have to see how the case plays out in court. This would be the first documented death associated with the "Whole 30" if the family prevails.
I hope they win. Regardless of the "diet", the driver shouldn't have been driving if he was impaired. If you are on one of those crazy fad diets, then for heaven's sake, take some of your special food with you, if you can't get it at restaurants or C-Stores. I have zeto sympathy for the driver, but a lot for the family of the dead woman.
I stumbled on a website today touting this diet and a couple of others. Fad diets are just more woo designed to sell books to gullible people looking for a quick fix. Moderation in all things.
No, he was negligent. He wasn't "hungry". He'd starved himself to the point where his blood sugar dropped dramatically, literally passing out behind the wheel, which wouldn't have happened but for his choice to adhere to a restrictive diet and skip meals.
I'd love to be a fly on the wall at his trial.
Impaired is impaired, whether you're drunk, stoned, took cold (or allergy) meds, or your blood sugar is too low. It's your responsibility to know where your head's at, and not drive if you're not completely sober.
Or, if you insist on trying a fad diet, break your diet for one day due to extenuating circumstances and start back on it tomorrow.
A few years ago a colleague of mine died after having a heart attack at the wheel of his car. He was on the Autobahn and managed to steer the car onto the break down strip before passing out and dieing.
If you pass out due to a medical problem you have no control about there is no negligence.
If you have certain medical disorders you're barred from driving.
If you inflict a medical condition on yourself, like starving yourself to the point of passing out, getting behind the wheel is negligent and should be treated like drunk driving. Because it's exactly the same thing.
"Them that does the hitting does the paying".
Yep, "impaired" and voluntarily so, would be my argument against him. Kinda doubt he'll qualify for accident or liability insurance for a while.. like, a lifetime or so….
The Whole30 does not advocate not eating/skipping meals. This really has nothing to do with the Whole30 — fruit, veggies, a hard boiled egg are all compliant and pretty readily available, even while traveling. It was an unfortunate choice not to eat that led to unfortunate, dire consequences.
I agree. Of course, someone who's into a fad diet (I don't know about this specific one), might skip the necessary hydration as well.
His choice not to eat was a direct result of following a "diet" that was too restrictive.
Everyone is different. That you can fast for days without issue does not mean that everyone else can.
Perhaps not everyone does. But most people could. It's ludicrous to assume that missing a single meal or even a few will result in this type of impairment in a healthy, non-diabetic individual. The human race would never have survived if we became gibbering idiots every time we hadn't eaten in more than 3 hours. The fact is, our hormonal systems are more than capable of mobilizing energy from our fat stores if we go without food. That's the whole point!! My spare tire wasn't put there for decoration!
If we're used to eating every four to six hours, then suddenly fasting will be a shock.


text:
\section{Introduction}

The Laplace-Beltrami problem along surfaces in three dimensions
appears in many fields, including plasma physics~\cite{beltrami3},
fluid mechanics~\cite{rahimian2010petascale,veerapaneni_2011},
electromagnetics~\cite{gendeb1,gendeb2,superconductor1,nedelec},
surface diffusion, pattern
formation~\cite{kim2020pattern,plaza2004effect}, and computational
geometry~\cite{kromer2018highly}.
In general, the Laplace-Beltrami operator is a 2nd-order variable coefficient
elliptic operator whose behavior is highly dependent on the nature of
the surface on which it is defined. Its eigenfunctions are useful in
describing band-limited functions defined along the associated
manifold or surface, and more generally, the related harmonic vector
fields provide a basis for important classes of functions appearing in
electromagnetics (e.g. non-radiating currents, etc.).

To this end, suppose that $\Gamma$ is at least a twice-differentiable
surface without boundary embedded in~$\mathbb{R}^{3}$. Let
$\nabla_{\Gamma}$ denote the intrinsic surface gradient, and let its
adjoint acting on the space of continuously differentiable
$\mathbb{L}^2$ tangential vector fields on the boundary, denoted by
$\nabla_{\Gamma} \cdot$, be the intrinsic surface divergence. Given a
mean-zero function $f$ defined along~$\Gamma$, the Laplace-Beltrami
problem is to find a mean-zero function $u$ satisfying
$\Delta_{\Gamma} u = f$, where $\Delta_{\Gamma}$ is the
Laplace-Beltrami operator defined as $\Delta_{\Gamma}= \nabla_{\Gamma}
\cdot \nabla_{\Gamma}$. The mean-zero condition on~$f$ and $u$ is
required for unique solvability due to a rank-deficiency of the
Laplace-Beltrami operator~\cite{oneil2018,gendeb1,nedelec}.

In many of the applications discussed above, the Laplace-Beltrami
problem naturally arises for the computation of the Hodge
decomposition of tangential vector fields. Any smooth tangential
vector field $\bF$ on the surface $\Gamma$ can be written as
\begin{equation}
\bF = \nabla_{\Gamma} \alpha + \bn \times \nabla_{\Gamma} \beta + \bH \, ,
\end{equation}
where $\alpha$, $\beta$ are smooth functions on $\Gamma$, $\bH$ is a
harmonic field satisfying $\nabla_{\Gamma} \cdot \bH = \nabla_{\Gamma}
\cdot \bn \times \bH = 0$, and $\bn$ is the unit normal to the surface
$\Gamma$.  The curl free component $\nabla_{\Gamma} \alpha$ and the
divergence free component $\bn \times \nabla_{\Gamma} \beta$ can be
computed via the solution of the following two Laplace-Beltrami
problems:
\begin{equation}
\Delta_{\Gamma} \alpha = \nabla_{\Gamma} \cdot \bF \, , \quad
\Delta_{\Gamma} \beta = - \nabla_{\Gamma} \cdot (\bn \times \bF) \, .
\end{equation}
A related problem is that of determining a basis for harmonic vector
fields on non-contractible surfaces, which plays a crucial role for
several integral representations in computational electromagnetics.
On a genus $g$ object, the space of harmonic vector fields forms a
$2g$-dimensional vector space. One can compute elements of this basis
via the solution of the two Laplace-Beltrami problems above for a
randomly generated smooth vector field $\bF$, which with high
probability tends to have a non-trivial projection onto the subspace
of harmonic
vector fields.

Numerical methods for the Laplace-Beltrami problem can be typically
classified into two categories.  The first category of methods rely on
the direct discretization of the differential operator on surface.
Examples include finite element
methods~\cite{bansch2005finite,bonito2020finite,burman2017cut,demlow2007adaptive},
virtual element
methods~\cite{beirao2014hitchhiker,frittelli2018virtual}, differencing
methods~\cite{wang2018modified}, and level set
methods~\cite{bertalmio2001variational,chen2015closest,greer2006fourth,macdonald2008level,macdonald2010implicit}.
The level set methods differ from the rest of the direct
discretizations of the PDE as they rely on using an embedded finite
difference grid in the volume~$\mathbb{R}^3$ to discretize a
thickening of the surface.

On the other hand, the other general category of methods is to
reformulate the Laplace-Beltrami problem as an integral equation along
the surface. For example, on a flat surface the Laplace-Beltrami
operator reduces to the standard Laplace operator and therefore
the Green's function is known to be~$G(\bx) = -\log{|\bx|}/(2\pi)$.
This Green's function~$G$ can be used
as a parametrix for the fundamental solution to the Laplace-Beltrami
operator on general surfaces as
well~\cite{kropinski_2014_fast}.  This idea has also been used for the
solution of the related Yukawa-Beltrami problem on general surfaces,
including surfaces with boundary
in~\cite{kropinski_2016_integral}. A separation of variables approach,
combined with an integral reformulation of an ODE, is presented
in~\cite{goodwill2021numerical}; this approach was also shown to be applicable in
geometries with edges. An alternate integral equation
based approach is to use the Green's function of the free-space
Laplace equation in three dimensions as either left or right
preconditioners for the Laplace-Beltrami problem~\cite{oneil2018}.

In this work, we present a fast multipole method (FMM) accelerated iterative
solver for the solution of the Laplace-Beltrami problems on complex
three-dimensional surfaces based on the integral formulation presented
in~\cite{oneil2018}, and using the locally corrected quadrature
approach of~\cite{greengard2020fmm}.  In some respects, this work can
be considered \emph{Part II} of the earlier work~\cite{oneil2018}.
The primary motivation of choosing this approach is two-fold: (1)
the resulting integral equations tend to be as well-conditioned as the
underlying physical problem (which is quite well-conditioned, in an
absolute sense),
and (2) since the representation
uses layer potentials related to the free space Green's function in
three dimensions, the numerical solver can be coupled to existing
optimized FMM libraries~\cite{greengard-1997,
  greengard2020fmm}.  We demonstrate the well-conditioned nature of
the integral equation through a battery of numerical experiments, each
of which incorporates a high-order representation of the surface and
high-order quadratures.  We also show the importance of what we refer
to as \emph{numerical
second-kindness} when discretizing the integral equation using locally
corrected quadrature methods. These methods rely on the accurate computation of
the principal-value part of the layer potentials in the
representation.  In particular, while there are several equivalent
integral formulations of the same PDE on the surface,
we show that the formulation which explicitly handles the identity
term of the integral equation tends to be the most stable (from a numerical
point of view).  Lastly, with the acceleration provided by FMMs,
our integral formulation can be used to compute a
basis for harmonic vector fields in relatively high-genus geometries.

The rest of the paper is organized in follows. In~\cref{sec:lapbel},
we describe the Laplace-Beltrami problem.  In~\cref{sec:inteq}, we
review the integral equation formulation presented
in~\cite{oneil2018}, and in~\cref{sec:solver}, we review a high order
FMM-accelerated iterative solver. In~\cref{sec:harm-vec}, we discuss
an application of the solver to the computation of a basis for
harmonic vector fields on non-contractible
surfaces. In~\cref{sec:num}, we present several numerical examples to
demonstrate the efficiency of our approach, including a discussion on
the importance of numerical second-kindness. Finally,
in~\cref{sec:conclusions}, we conclude by discussing avenues for
future research.

\section{The Laplace-Beltrami problem \label{sec:lapbel}}
The Laplace-Beltrami operator, also known as the surface Laplacian, is
the generalization of the Laplace operator to general (smooth)
manifolds.  In this work, we will focus on smooth surfaces embedded in
three dimensions.  Suppose $\Gamma \subset \mathbb{R}^{3}$ is an
oriented surface with at least two continuous derivatives, and let
$\bx(u,v): D \subset \mathbb{R}^{2} \to \mathbb{R}^{3}$ denote a local
parametrization of~$\Gamma$. The first fundamental form, or
equivalently the metric tensor at $\bx(u,v)$, is given by
\begin{equation}
    g = \begin{bmatrix}
    g_{11}       & g_{12}  \\
    g_{21}       & g_{22}
\end{bmatrix}
= \begin{bmatrix}
    \xu \cdot \xu       &  \xu \cdot \xv \\
    \xv \cdot \xu     & \xv \cdot \xv
\end{bmatrix} ,
\end{equation}
where $\xu$ and $\xv$ are abbreviations for  $\frac{\partial \bx}{\partial u}$ and $\frac{\partial \bx}{\partial v}$ respectively and are the tangents to the surface at $\bx \in \Gamma$.
The unit normal at $\bx$ can written as
\begin{equation}
  \bn(\bx) = \frac{\xu \times \xv}{ | \xu \times \xv |}.
\end{equation}
The unit vectors in the directions of $\xu$, $\xv$, and $\bn(\bx)$ form
a local coordinate system (not necessarily orthogonal) at each
point~$\bx\in \Gamma$.

Next, suppose that $\boldsymbol{F}$ is a smooth tangential vector
field defined along~$\Gamma$ and is expressed in terms of the local
tangent vectors $\xu$, $\xv$ as
 \begin{equation}
 \boldsymbol{F}(u,v) = F_{1} \xu + F_{2} \xv \, .
 \end{equation}
The surface divergence of $\boldsymbol{F}$, denoted by
$\nabla_{\Gamma} \cdot \boldsymbol{F}$ is then~\cite{nedelec,frankel} given by
 \begin{equation}
 \nabla_{\Gamma} \cdot \boldsymbol{F} = \frac{1}{\sqrt{|g|}} \left(
 \partial_{u} \left( \sqrt{|g|} F_{1} \right) + \partial_{v} \left( \sqrt{|g|} F_{2} \right)
 \right)  \, ,
 \end{equation}
 where $|g|$ is the determinant of the metric tensor $g$.  The surface
 gradient operator $\nabla_{\Gamma}$, formally defined as the adjoint of the surface divergence operator on the space
 of~$\mathbb{L}^{2}$ functions with respect to the induced metric on $\Gamma$,
 is given by
 \begin{equation}
 \nabla_{\Gamma} \psi = \left( g^{11} \partial_{u} \psi + g^{12} \partial_{v} \psi \right) \xu +
 \left( g^{21} \partial_{u} \psi + g^{22} \partial_{v} \psi \right) \xv \, ,
 \end{equation}
 where $\psi = \psi(u,v)$ is a scalar function defined on~$\Gamma$
 and $g^{ij}$ denote the components of the inverse of the
 metric tensor, denoted by $g^{-1}$.

The Laplace-Beltrami operator is then defined as~$\surflap =
\nabla_{\Gamma} \cdot \nabla_{\Gamma}$, and is explicitly given by
the expression
\begin{equation}
     \Delta_{\Gamma} \psi = \frac{1}{\sqrt{|g|}} \left( \partial_{u}
     \left[ \sqrt{|g|} \left(g^{11} \partial_{u} \psi + g^{12}
       \partial_{v} \psi \right) \right] + \partial_{v} \left[
       \sqrt{|g|} \left(g^{21} \partial_{u} \psi + g^{22} \partial_{v}
       \psi \right) \right] \right).
\end{equation}
The standard Laplace-Beltrami problem is to solve the following
PDE for~$\psi$ along~$\Gamma$,
\begin{equation}
  \label{eq:lapbel}
  \surflap \psi = f,
\end{equation}
where~$f$ is a given function. Conditions on the regularity required
in~$f$, and the subsequent regularity obtained in~$\psi$ have been
studied in~\cite{rosenberg, jost,epstein-2010, goodwill2021numerical}.
We will assume that all functions and surfaces are
sufficiently smooth so as to obtain high order accuracy in the
subsequently numerical examples.

As is clear, the Laplace-Beltrami operator maps constant functions
along~$\Gamma$ to zero. Thus, the equation~\eqref{eq:lapbel} is
rank-one deficient~\cite{imbertgerard_2017, sifuentes_2015} (and
formally self-adjoint). The
following lemma~\cite{rosenberg, oneil2018} summarizes a classical
result regarding the well-posedness of the above Laplace-Beltrami
problem.

\begin{lemma}
 \label{lem:meanzero}
Let $\Gamma$ be a smooth, closed, and orientable
boundary. Along~$\Gamma$, the
Laplace-Beltrami operator is uniquely invertible as a map from
$\mathbb{M}_{0} \to \mathbb{M}_{0}$, the space of mean-zero functions
defined on $\Gamma$, \,
\begin{equation}
\mathbb{M}_{0} = \left\{ \psi \, : \, \int_{\Gamma} \psi \, da = 0 \right\} \, .
\end{equation}
Indeed, if the function~$f$ defined along~$\Gamma$ has mean zero, then
there exists a unique mean-zero solution,~$\psi$, to the
Laplace-Beltrami problem~$\surflap \psi = f$.
\end{lemma}

\section{Integral formulation \label{sec:inteq}}
In this section, we review an equivalent
well-conditioned second-kind integral equation for the solution of the
Laplace-Beltrami problem. This formulation relies on preconditioning
the Laplace-Beltrami operator on both the left and right using the
Laplace single layer potential.  The resulting Fredholm operator can
be expressed in terms of standard layer potentials for the free space
Laplace equation in three dimensions.

Before describing this
formulation, we first define the relevant layer
potentials used in the formulation.
The Green's function $G$
for Laplace's equation in
three dimensions given by
\begin{equation}
G(\bx,\by) = \frac{1}{4\pi |\bx-\by| }  \, .
\end{equation}
The  single and double layer potentials are then given by:
\begin{equation}
\mathcal{S}[\sigma](\bx) = \int_{\Gamma} G(\bx,\by) \, \sigma(\by) \, da \,  , \quad
\mathcal{D}[\mu](\bx) = \int_{\Gamma} \left( \nabla_{\by} G(\bx,\by)
\cdot \bn(\by) \right)
\mu(\by) \, da.
\end{equation}
When the \emph{target} point $\bx$ lies on~$\Gamma$, the integral
defining the double layer should be
interpreted in a principal value sense. For smooth surfaces $\Gamma$,
both $\mathcal{S}$ and $\mathcal{D}$ are compact operators as maps
from~$\mathbb{L}^{2}(\Gamma) \to \mathbb{L}^{2}(\Gamma)$.
 Related to the single and double layer
potentials are the restrictions of their normal derivatives to
$\Gamma$ given by
\begin{equation}
  \begin{aligned}
    \mathcal{S}'[\sigma](\bx) &= \bn(\bx) \cdot \nabla \int_{\Gamma}
    G(\bx,\by) \sigma(\by) \, da, \\
    \mathcal{D}'[\mu](\bx) &= \int_{\Gamma} \bn(\bx) \cdot
    \nabla_{\bx} \nabla_{\by}  G(\bx,\by) \cdot \bn(\by) \mu(\by)
    \, da.
  \end{aligned}
\end{equation}
The integral defining $\mathcal{S}'$ is to be interpreted in a
principal value sense, and that for $\mathcal{D}'$ is to be
interpreted in a finite-part sense~\cite{kress_2014, nedelec}. The
operator $\mathcal{S}'$ is also a compact operator on
$\mathbb{L}^{2}(\Gamma)$, whereas~$\mathcal D'$ is a hypersingular
operator.

Finally, let $\mathcal{S}''$ denote the restriction of the second
normal derivative of $\mathcal{S}$ to $\Gamma$ given by:
\begin{equation}
  \mathcal{S}''[\sigma](\bx) = \int_{\Gamma} \left(
  \bn(\bx) \cdot \nabla_{\bx} \nabla_{\bx}  G(\bx,\by) \cdot \bn(\bx)
  \right) \sigma(\by) \, da,
\end{equation}
where, as for the integral defining $\mathcal{D}'$, the integral
should be interpreted in a finite-part sense \cite{kress_2014,
  nedelec, oneil2018}. It can be shown that the \emph{difference}
operator $\mathcal{S}'' + \mathcal{D}'$ is a compact operator
on~$\mathbb{L}^{2}(\Gamma)$~\cite{nedelec}, as the dominant
singularities in the integrand cancel each other out.

In the following theorem, we now review the equivalent second-kind
boundary integral equation presented in~\cite{oneil2018}, for the
solution of the Laplace-Beltrami equation \eqref{eq:lapbel}.
\begin{theorem}
Suppose that $f$ is a mean-zero function in
$\mathbb{L}^{2}(\Gamma)$. Then there exists a unique solution~$\sigma$
in $\mathbb{L}^{2}(\Gamma)$ which satisfies
\begin{equation}
  \label{eq:bie1}
-\frac{\sigma}{4} + \mathcal{D}^2[\sigma] -
\mathcal{S}\left(\mathcal{S}'' + \mathcal{D}' -2H\mathcal{S}'
\right)[\sigma] + \mathcal{SWS}[\sigma] = \mathcal{S}[f] \, ,
\end{equation}
and $\int_{\Gamma} \mathcal{S}[\sigma] da = 0$.  Here $H$ is the mean
curvature along~$\Gamma$, and $\mathcal{W}$ is the operator given by
\begin{equation}
\mathcal{W}[\sigma] = \frac{1}{|\Gamma|} \int_{\Gamma} \sigma \, da \,.
\end{equation}
Furthermore, $\psi = \mathcal{S}[\sigma]$ is the unique mean-zero
solution to $\Delta_{\Gamma} \psi = f$.
\end{theorem}
A proof of this result can be found in~\cite{oneil2018}.

\section{An FMM-accelerated solver \label{sec:solver}}
\label{sec:fmmsolver}

In this section, we briefly describe the details of our numerical solver for
integral equation formulations of the Laplace-Beltrami problem. A full detailed discussion would merely be a
reproduction of much of the content of~\cite{greengard2020fmm}, which
presents a FMM-accelerated library of solvers for various
boundary integral equations. Our numerical solver is based on the
software packages~\texttt{fmm3dbie}\cite{FMM3DBIE} and~\texttt{FMM3D}~\cite{FMM3D};
subsequently, examples for the Laplace-Beltrami problem
  will be included in the package \texttt{fmm3dbie}. We now give a
very brief overview of the solver, and make a few remarks which are
specific to the Laplace-Beltrami problem.

For simplicity, consider a second-kind integral equation of the form
\begin{equation}
\label{eq:skie-demo}
\sigma(\bx) + \int_{\Gamma} K(\bx,\bx') \, \sigma(\bx') \, da(\bx') = f(\bx) \,
\quad \bx \in \Gamma \,,
\end{equation}
where $K(\bx,\bx')$ is either $G(\bx,\bx')$ or one it's directional derivatives.
Suppose that the surface $\Gamma$ is represented via a disjoint union of patches $\Gamma_{j}$, $\Gamma = \cup_{j=1}^{\Npat} \Gamma_{j}$ and that each patch $\Gamma_{j}$ parametrized by a non-degenerate chart $\bX^{j} : T_{0} \to \Gamma_{j}$, where
$T_{0}$ is the standard simplex given by
\begin{equation*}
T_{0} = \{ (u,v): u \geq 0 \,, v\geq 0\, , u+v \leq 1 \} \subset \mathbb{R}^{2} \, .
\end{equation*}
An order $p$ discretization of $\Gamma$ is one where the components of $\bX^{j}, \partial_{u} \bX^{j}$, and $\partial_{v} \bX^{j}$ are represented  by a polynomial expansion of total degree less than $p$ on $T_{0}$.
One way to obtain such a representation is to sample the chart and the derivative information at order $p$ Vioreanu-Rokhlin nodes~\cite{vioreanu_2014} which are stable interpolation nodes for orthogonal polynomials~\cite{bremer_2012, koornwinder_1975} on $T_{0}$.

\begin{remark}
Triangulations of surfaces obtained from CAD or standard meshing
packages often tend to be low order and introduce several artificial
edges and corners on the discretized surface. In our examples, with
the aim of showing high-order convergence, we use the surface
smoothing algorithm discussed in~\cite{vico2020surface} to convert these
low-order triangulations of complex geometries to arbitrarily high
order triangulations of a \emph{nearby} related~$C^{\infty}$
surface. In some of our examples, analytic parameterizations of the
domains are known and are used instead of the algorithm
of~\cite{vico2020surface}. The algorithm was only applied to complex
geometries, such as the multi-holed object in Figure~\ref{fig:g10}.
\end{remark}

Let $ \{ \bx_{i} \}_{i=1}^{N}$ denote the union of the samples of the charts $\bX^{j}$ at the Vioreanu-Rokhlin nodes
of order $p$.
A Nystr\"{o}m discretization of~\cref{eq:skie-demo} is given by
\begin{equation}
\label{eq:nys}
\sigma_{i} + w_{ii} \sigma_{i} + \sum_{j\neq i} w_{ij} K(\bx_{i},\bx_{j}) \sigma_{j} = f(\bx_{i}) \, ,
\end{equation}
where $\sigma_{i}$ is an approximation for the solution $\sigma(\bx_{i})$.
In~\cite{greengard2020fmm}, the layer potential $\int_{\Gamma}
K(\bx_{i},\bx') \, \sigma(\bx') \, da(\bx')$ for $\bx_{i} \in \Gamma_{\ell}$
is approximated using a combination of generalized Gaussian quadrature for the contribution of $\Gamma_{\ell}$ to the layer potential, adaptive integration for patches $\Gamma_{j}$ which are close to $\bx_{i}$ and target independent oversampled quadratures for all the remaining patches. Let  $\{ \bs_{i}, w_{i} \}_{i=1}^{\Nover}$ denote the oversampled quadrature nodes and weights for integrating smooth functions on $\Gamma$, then the Nystr\"{o}m discretization in~\cref{eq:nys} takes the form
\begin{equation}
\label{eq:nys-lcq}
\sigma_{i} + w_{ii} \sigma_{i} + \sum_{j=1}^{n_{i}} w_{ij} \, K(\bx_{i},
\by_{ij}) \, \tilde{\sigma}_{ij} + \sum_{\substack{j=1 \\ \bs_{j} \in
    \textrm{Far}(\bx_{i})}}^{\Nover} w_{j} \, K(\bx_{i}, \bs_{j}) \,
\tilde{\sigma}_{j} = f(\bx_{i}) \, ,
\end{equation}
where $\tilde{\sigma_{j}}$ and $\tilde{\sigma}_{ij}$ are the
interpolated values of $\sigma(\bx_{i})$ at $\bs_{j}$ and $\by_{ij}$,
respectively, obtained via local polynomial interpolation of the
samples $\sigma_{i}$, and $\by_{ij}, w_{ij}$ are the quadrature nodes
and weights (which can be precomputed and stored) used in the adaptive
integration procedure to accurately compute the contribution of the
layer potential due to self panel $\Gamma_{\ell}$ and the near panels
at the target location $\bx_{i}$. There are $O(1)$ patches which are
in the near field of each target, and since the far field part of the
layer potential is computed using a target independent quadrature
rule, the corresponding contribution can be computed using standard
fast multipole methods~\cite{greengard-1997,
  greengard2020fmm}. If~\cref{eq:nys-lcq} is well-conditioned, then we
can obtain the solution $\sigma_{i}$ in $O(N)$ time using an iterative
algorithm such as GMRES~\cite{saad-1986}.

In the context of solving the Laplace-Beltrami problem, there are two
details that aren't already a part of the
\texttt{fmm3dbie}~\cite{FMM3DBIE} package: (1) the contribution of
$\mathcal S'' + \mathcal D'$ needs to be handled as a single operator since both
$\mathcal S''$ nor $\mathcal D'$ are hypersingular operators, but their sum $\mathcal S'' +
\mathcal D'$ is a compact operator of order $-1$, and (2) the mean curvature
$H$ needs to be computed by spectral differentiation of $\partial_{u}
\bX^{j}$ and $\partial_{v} \bX^{j}$.

\section{Harmonic vector fields \label{sec:harm-vec}}
For a surface $\Gamma$ with unit normal $\bn$, a harmonic vector field $\bH$ is one which satisfies
\begin{equation}
\nabla_{\Gamma} \cdot \bH = 0 \, , \quad \nabla_{\Gamma} \cdot (\bn
\times \bH) = 0 \, .
\end{equation}
On a genus $g$ surface, it is well known that the space of harmonic
vector fields is $2g$ dimensional~\cite{dai_2014, epstein-2013,
  epstein-2010}. Moreover, it follows from the definition that if
$\bH$ is a harmonic vector field, then $\bn \times \bH$ is also a
harmonic vector field which is linearly independent of $\bH$.  Any
tangential $C^{1}$ vector field $\bF$ along a smooth surface admits a
Hodge decomposition~\cite{epstein-2010, epstein-2013}:
\begin{equation}
\bF = \nabla_{\Gamma} \alpha + \bn \times \nabla_{\Gamma} \beta + \bH \, ,
\end{equation}
where $\alpha$ and $\beta$ are scalar functions defined on the surface
$\Gamma$ and $\bH$ is a harmonic vector field.  Given a smooth
tangential vector field $\bF$, one can use the existence of
this decomposition to
compute linearly independent harmonic vector fields $\bH$ by solving
the following Laplace-Beltrami equations and subsequently computing $\bH$:
\begin{equation}
\begin{aligned}
\Delta_{\Gamma} \alpha = \nabla_{\Gamma} \cdot \bF \, , &\quad \Delta_{\Gamma} \alpha = -\nabla_{\Gamma} \cdot (\bn \times \bF),  \\
\bH = \bF - &\nabla_{\Gamma} \alpha - \bn\times \nabla_{\Gamma} \beta \, .
\end{aligned}
\end{equation}

A convenient way of choosing such smooth vector fields $\bF$ is to
first define $\bV \in \bR^{3}$ where each component is a low degree random
polynomial, and then set $\bF = \bn \times \bn \times \bV$.
The maximum degree of Legendre polynomials should be at least $\lceil (2g)^{1/3} \rceil$ so that dimension of the space of polynomials is greater than
the dimension of the space of the vector fields. For the examples in this paper, we choose
the degree to be $\lceil (2g)^{1/3} \rceil + 3$.
Moreover, suppose that
$\Gamma$ is contained in $(-L,L)^3$, then each component is defined to
be a scaled tensor product Legendre polynomial~\cite{olver2010}
$P_{\ell}(x/2L) P_{m} (y/2L) P_{n} (z/2L)$, where $P_{n}(x)$ is the
standard Legendre polynomials of degree $n$ on $[-1,1]$.  The extra
factor of $2$ in the choice of Legendre polynomials is to avoid
requiring additional degrees of freedom to represent $\bF$ on the
surface due to large gradients of the Legendre polynomials near the
end points $-1$ and $1$.

With high probability, different random polynomial vector fields $\bF$
result in different projections on the different elements of the
$2g$-dimensional space of vector fields. It is easy to show that if
$\bH$ is a harmonic vector field then $\bn \times \bH$ is also a
harmonic vector field which is linearly independent of $\bH$.  Thus,
in practice one can construct a basis for the harmonic vector field
subspace by computing $\bH$ for $g$ different vector fields $\bF$, and
the remaining $g$ vector fields of the basis are defined as $\bn
\times \bH$.

\section{Numerical examples \label{sec:num}}

In this section, we provide several numerical examples demonstrating
the accuracy and computational efficiency of our solver for the
Laplace-Beltrami problem, as well as an application of the solver for
computing harmonic vector fields.

\subsection{Numerical second-kindness}
We first turn our attention to the differences in numerical
discretizations of the following three (mathematically equivalent)
formulations of an integral equation for the Laplace-Beltrami PDE:
\begin{subequations}
  \label{eq:reps}
  \begin{align}
    \mathcal S \Delta_{\Gamma} \mathcal S &= \left( \mathcal S \nabla_\Gamma \cdot \right) \left( \nabla_\Gamma \mathcal S
    \right) \label{rep1} \\
    &= \left( \nabla_\Gamma \cdot \mathcal S \right) \left( \nabla_{\Gamma} \mathcal S \right) \label{rep2} \\
    &= -\frac{1}{4} \mathcal I
    + \mathcal{D}^2 - \mathcal{S}\left(\mathcal{S}'' + \mathcal{D}' -2H\mathcal{S}' \right) \label{rep3} \, .
  \end{align}
\end{subequations}
For convenience of exposition, we have dropped the rank-one operator
$\mathcal W$ as it is not relevant to the discussion in this section.  While
the above operators (Eq. (\ref{rep1}), (\ref{rep2}), (\ref{rep3})) are
all equal as maps on smooth functions on~$\Gamma$, their performance
when discretized numerically differs significantly on patch-based
discretizations of the surface.

The first approach in~\cref{rep1} of composing $\mathcal S \nabla_{\Gamma}
\cdot$ with $\nabla_{\Gamma} \mathcal S$ is extremely unstable on patch-based
discretizations where smooth functions are represented as independent
polynomial expansion on each patch.  The discretization does not
enforce any smoothness constraint on functions across the patch
interfaces which results in a spurious null space proportional to the
number of degrees of freedom used to discretize the problem.  In
particular, the surface divergence is computed after projecting the
values of $\nabla_{\Gamma} \mathcal S$ onto a truncated Koornwinder
polynomial expansion. On each patch, the surface divergence operator
has a null space proportional to the order of the discretization. To
circumvent this issue, one could use discretizations commonly used in
finite element methods which preserve smoothness of functions across
patches.~\Cref{rep2} is an equivalent formulation which avoids this
issue.

Even though~\cref{rep2} avoids the spurious null-space issue,
both~\cref{rep1,rep2} tend to perform poorly when using iterative
solver (as compared to using~\cref{rep3}).  One of the key reasons for
this difference is that \emph{any} numerical discretization of
integral operators is inherently compact since the operator is
approximated via a finite-rank one. This results in the spectrum of
the discretized integral operator clustering closely to the origin,
resulting in numerical conditioning issues. Explicitly adding the
identity term, as is the case in~\cref{rep3}, results in the spectrum
of the discretized integral operator clustering at $-1/4$. This
\emph{greatly} enhances the convergence of iterative methods such as
GMRES~\cite{saad-1986}.
These differences in behavior can be demonstrated by applying each of
the operators on the right-hand sides of~\cref{eq:reps} to a function
whose Laplace-Beltrami signature is known explicitly. On the unit
sphere, for a function~$u$ given in terms of its spherical harmonic
expansion
\begin{equation}
u = \sum_{n=1}^{2} \sum_{m=0}^{n} c_{nm} \, Y_{nm}(\theta,\phi),
\end{equation}
where $Y_{nm}(\theta,\phi)$ are the spherical
harmonics of degree $n$ and order $m$,
the action of the operator $\mathcal S(\Delta_{\Gamma} + \mathcal W) \mathcal S$
is diagonal and is given as~\cite{vico_2014}
\begin{equation}
  \begin{aligned}
    v &= \mathcal S(\Delta_{\Gamma} + \mathcal W) \mathcal S [u] \\
    &= -\sum_{n=1}^{2}
    \sum_{m=0}^{n} c_{nm} \frac{ n(n+1)}{(2n+1)^2} Y_{nm}(\theta,\phi).
    \end{aligned}
\end{equation}
We discretize the layer potentials in representations
in~\cref{rep1,rep2,rep3} on a unit sphere  based on the procedure discussed
in~\cref{sec:fmmsolver}; the unit sphere is discretized
using~$\Npat$ number of
patches, each of order
$p$. We then  apply these operators to a function~$u$ of
the form above.  We verify the correctness of our discretization by
computing the~$\mathbb{L}^2$ error in the result (compared to the
known solution~$v$ above).
The coefficients $c_{nm}$, $n=1,2$, $0\leq m \leq n$ are chosen uniformly
from the interval $(-0.5,0.5)$.
This error is reported
as~$\varepsilon_{mv}$ in~\cref{tab-comp}.  For all three
representations, we note that the error converges to $0$ at the
expected rate of $O(h^{p-1})$.

\begin{table}[b!]
\centering
\begin{tabular}{cccSS} \hline
     {Representation} & {$\Npat$} & {$N$} & {$\varepsilon_{mv}$  } & {$\varepsilon_{s}$}   \\ \hline
    \multirow{3}{*}{\cref{rep1}} & 48  & 480   & 3.1E-2 & 1.1E+12 \\
    & 192 & 1920 & 6.8E-3 & 4.6E+10 \\
    & 768 & 7680 & 9.0E-4 & 3.0E+7 \\ \hline
        \multirow{3}{*}{\cref{rep2}} & 48  & 480   & 1.6E-2 & 2.2E-1 \\
    & 192 & 1920 & 1.4E-3 & 3.7E-2 \\
    & 768 & 7680 & 1.8E-4 & 2.3E-3 \\ \hline
        \multirow{3}{*}{\cref{rep3}} & 48  & 480   & 8.5E-3 & 8.1E-3 \\
    & 192 & 1920 & 6.3E-4 & 6.1E-4 \\
    & 768 & 7680 & 8.1E-5 & 8.0E-5 \\ \hline
\end{tabular}
\caption{ Comparison of error in applying $\mathcal S(\Delta_{\Gamma} +
  \mathcal W)\mathcal S$ (denoted by $\varepsilon_{mv})$ and inverting
  $\Delta_{\Gamma} + \mathcal W$ (denoted by $\varepsilon_{s})$ for the three
  equivalent representations given in~\cref{rep1,rep2,rep3} on the
  unit sphere. } \label{tab-comp}
\end{table}

Next, in order to study the relative performance of these
representation in iterative solvers, we compare the spectral
properties of the three discretized linear systems.  We form the dense
matrices for $\Npat=192$ and compute their singular value
decompositions.  The resulting singular values are plotted
in~\cref{fig1}. As expected, we see that~\cref{rep1} has a large
spurious null space and that the other two discretizations are full
rank. Moreover, the singular values of~\cref{rep2} cluster at $0$,
while the singular values of~\cref{rep3} cluster at 0.25.  We then
solve the Laplace-Beltrami problem~$(\Delta_\Gamma + \mathcal W)\psi = u$
using each of these discretized linear systems and GMRES.  On the
right in~\cref{fig1}, we plot the estimated residual as a function of
iteration number for the three representations. The GMRES residual
for~\cref{rep1,rep2} stagnates at the level of discretization error,
while~\cref{rep3} is more robust and the residual converges to machine
precision independent of discretization error.  The analytical
  solution for the data $u$ expressed as a spherical harmonic
  expansions is known to be
  \begin{equation}
    \begin{aligned}
      \psi &= (\Delta_{\Gamma} + \mathcal W)^{-1} u \\
      &= -\sum_{n=1}^{2} \sum_{m=0}^{n}
      \frac{c_{nm}}{n(n+1)} Y_{nm}(\theta,\phi).
      \end{aligned}
\end{equation}
In~\cref{tab-comp}, we report the $\mathbb{L}^{2}$ error in the
computed solution $\psi$, denoted by
$\varepsilon_{s}$. For~\cref{rep2,rep3}, the error converges to $0$ at
the expected rate of $O(h^{p-1})$ (even though GMRES stagnates at
discretization error for~\cref{rep2}), while for~\cref{rep1}, the
spurious null space completely pollutes the solution.

\begin{figure}[t!]
\centering
\includegraphics[width=0.9\linewidth]{repcomp}
\caption{The singular values of the matrices for the three
  representations~\cref{rep1,rep2,rep3} for a sphere discretized with
  $\Npat=192$, and $p=4$ (left) and estimated GMRES residual as a
  function of iteration number (right). }
\label{fig1}
\end{figure}

\subsection{Order of convergence}
Next, we demonstrate the accuracy and convergence of our solver
following the representation in \cref{rep3}. To this end, we use the
solver to compute a numerical solution of Laplace-Beltrami equation on
a {torus} with major radius $R=2.25$ and minor radius $r=0.75$
and compare it with the known analytical solution. Suppose that the
torus is parameterized as
\begin{equation}
  \bX(s,t) =
  \begin{pmatrix}
    (R+r \cos{t}) \cos{s} \\
    (R+r\cos{t})\sin{s} \\
    r\sin{s}
  \end{pmatrix},
  \qquad \text{for } s,t \in [0,2\pi].
\end{equation}
A simple calculation shows that for this geometry,
\begin{equation}
  \label{eq:alpha}
  \begin{aligned}
(\Delta_{\Gamma} + \mathcal W) z &=  \frac{z}{r^2 \sqrt{x^2 + y^2}} \left( R-
    2\sqrt{x^2+y^2} \right) \\
    &= \alpha(x,y,z).
    \end{aligned}
\end{equation}
Next, suppose that the torus is discretized with $\Npat$ patches, each
of order
$p$, resulting in \mbox{$N = \Npat \cdot p (p+1)/2$} discretization
points. We compute the numerical solution~$\psi$ to the
  Laplace-Beltrami problem $(\Delta_{\Gamma}+\mathcal W) \psi = \alpha$,
  with~$\alpha$ defined above, on the
  torus~$\Gamma$.  In~\cref{tab:accuracy}, we tabulate the numerical
results by presenting the error in the computed numerical solution
(denoted by $\varepsilon_{s}$), the estimated rate of convergence
$\rho_{c}$,  and the mean of the computed solution (denoted by
$\psi_{m}$). We observe expected order convergence to the analytical
solution where the observed order is appropriately dependent on the
discretization order $p$ of the patches.

\begin{table}[t!]
\centering
\begin{tabular}{cccScS} \toprule
    {$p$} & {$\Npat$} & {$N$} & {$\varepsilon_{s}$} & {$\rho_{c}$} & {$\psi_{m}$}   \\ \midrule
    3  & 100  & 600 & 8.2E-3  &  & 1.8E-15 \\
    3  & 400  & 2400 & 1E-3  & 3.03 &  1.3E-12\\
    3  & 1600  & 9600 & 1.2E-4  & 3.05 &  1.3E-12\\ \midrule
    4  & 100   & 1000 & 1.1E-3   &  & 6.3E-13   \\
    4  & 400  & 4000 & 1.3E-4  & 3.08 & 1.1E-12   \\
    4  & 1600  & 16000 & 1.7E-5 & 2.94& 6.4E-14    \\   \midrule
    5  & 100   & 1500 & 6.3E-5   &  & 6.3E-11   \\
    5  & 400  & 6000 & 2E-6  & 4.97 & 1.6E-12   \\
    5  & 1600  & 24000 & 6.5E-8 & 4.94& 5.2E-13    \\   \midrule
    7  & 100  & 2800 & 3.3E-7  &  &  7.7E-13 \\
    7 & 400  & 11200 & 2.9E-9   & 6.83 & 1.0E-12 \\
    7 & 1600  & 44800 & 1.1E-10  & 4.72 &  5.5E-13  \\\midrule
    9  & 16  & 720 & 4.8E-6  &  &1.1E-14   \\
    9  & 64  & 2880 & 1.2E-8  & 8.64 & 9.8E-14   \\
    9  & 256  & 11520 & 1E-10  & 6.9 & 4.2E-13   \\  \bottomrule
\end{tabular}
\caption{Convergence results for $(\Delta_{\Gamma} + \mathcal W)\psi =
  \alpha$, with~$\alpha$ defined in~\cref{eq:alpha}, on a torus with
  major radius $R=2.25$ and minor radius
  $r=0.75$} \label{tab:accuracy}
\end{table}

\subsection{An alternative integral equation}

An alternative second-kind integral equation can be derived for the
Laplace-Beltrami problem merely by right preconditioning
by~$\mathcal S^2$. In short, if we let~$\psi = \mathcal S^2[\sigma]$, then after a
suitable application of various Calder\'on
identities~\cite{contopanagos-2002, oneil2018}, the Laplace-Beltrami
problem~$(\surflap + \mathcal W) \psi = f$ is replaced by
\begin{equation}
 \left( -\frac{\mathcal I}{4} + (\mathcal S')^2 - (\mathcal S''+\mathcal D' - 2H\mathcal S')\mathcal S+ \mathcal W \mathcal S^2
 \right) [\sigma] = f.\label{rep4}
\end{equation}
This formulation is also second-kind, but its compact part differs
ever so slightly from that of~\cref{rep3}. In~\cref{tab:comptor}, we
compare the two integral representations,~\cref{rep3}~and \cref{rep4},
with respect to the number of GMRES iterations required and the accuracy
in computing $(\Delta_{\Gamma} + \mathcal W)\psi = \alpha$ on the torus.  Since
the compact part of both the representations don't differ by much, the
numerical performance of both these representations is practically
indistinguishable.

\begin{table}[b!]
\begin{tabular}{ccc|cc|cc} \toprule
     {$\Npat$} & $p$ & {$N$} & {$\mathbb{L}^{2}$ error with
       $S\Delta_{\Gamma}S$  } & \# iterations  & {$\mathbb{L}^{2}$
       error with $\Delta_{\Gamma}S^{2}$}  & \# iterations   \\ \midrule
    100  & 5 &  1500   & 6.3 E-5 & 9 & 6.5 E-5 &  10  \\
    400  & 5 & 6000   & 2.0 E-6 & 8 & 2.0 E-6 &  8 \\
    1600  & 5 &24000 & 6.5 E-8 & 8 & 6.4 E-8 &  8   \\\bottomrule
\end{tabular}
\caption{ Comparison of GMRES iteration counts and errors
  for~$S(\Delta_{\Gamma} + \mathcal W)S$ and $(\Delta_{\Gamma} + \mathcal W)S^{2}$
  for a torus with major radius 2.25 and minor radius
  0.75.} \label{tab:comptor}
\end{table}

As can be seen from the results above, there is not any notable
difference in solving the Laplace-Beltrami PDE by representing the
solution as $\psi=\mathcal S[\sigma]$ and preconditioning on the left with another
application of $\mathcal S$, or by representing $\psi= \mathcal S^{2}[\sigma]$, both in
terms of the accuracy achieved and the performance of the iterative
solvers. However, various applications may dictate one representation
over the other.
In the context of solving Maxwell's equations or
static currents in type I superconductors using the Debye source
representation~\cite{superconductor1}, we find that using $\psi = \mathcal S^{2}
[\sigma]$ is more faithful to the spectral properties of the
solution~\cite{gendeb1,gendeb2,superconductor1}.  As a map from
mean~zero functions to themselves, if $\Delta_{\Gamma} \psi = f$ then $\psi
= \Delta_{\Gamma}^{-1} f$, which implies that the map from $f \to \psi$
is a pseudo-differential operator of order $-2$.  However, if we
represent $\psi=\mathcal S[\sigma]$, then the map from $\sigma \to \psi$ is a
pseudo-differential operator of order $-1$, while for the
representation $\psi = \mathcal S^2[\sigma]$, the map from $\sigma \to \psi$ is
indeed a pseudo-differential operator of order $-2$.  This
significantly affects the spectral properties of some of the operators
appearing in the generalized-Debye representation.

For example, one of the operators in the Generalized-Debye
representation~\cite{epstein-2010} is
\begin{equation}
\mathcal A[q] = \bn \cdot \nabla \times \mathcal S_{k} [\bn \times \nabla_{\Gamma}
  \Delta_{\Gamma}^{-1} q] \,,
\end{equation}
where $q$ is a mean-zero surface charge and $\mathcal S_{k}$ is the single-layer
potential for the Helmholtz equation~\cite{papas} with wave number $k$
given by
\begin{equation}
\mathcal S_{k}[\sigma] = \int_{\Gamma} \frac{e^{ik|\bx-\by|}}{4\pi |\bx-\by|}
\, \sigma(\by) \, da(\by) \, .
\end{equation}
Note that $\mathcal A$ is a pseudo-differential operator of order $-1$ when
acting on~$q$.
If we represent the quantity~$\Delta_{\Gamma}^{-1} q$ as
$\psi=\mathcal S[\sigma]$,
then we need to discretize the following operator instead of $\mathcal A$,
\begin{equation}
\tilde{\mathcal A}[\sigma] = \bn \cdot \nabla \times \mathcal S_{k} [\bn \times \nabla_{\Gamma} \mathcal S [\sigma]] \, .
\end{equation}
It can be shown that~$\tilde{\mathcal A}$ is a second-kind Fredholm operator
of the form $-\mathcal I/4 + \mathcal K$, for some compact operator $\mathcal K$, when
acting on~$\sigma$ (as opposed to $\mathcal A$ which is itself compact
operator).  To the best of our knowledge, explicit Calder\'on
identities are not known for this operator. However, one can deduce
the second-kindness of the operator and the strength of the identity
term by analyzing the action of the operator on spherical harmonics
when the boundary is the unit sphere.  In particular, if $\Gamma$ is
the unit sphere, then
\begin{equation}
\tilde{\mathcal A}[Y_{nm}] = -\frac{i k j_{n}(k) h_{n}(k) n (n+1)}{2n+1}  Y_{nm} = c_{n} Y_{nm} \, ,
\end{equation}
where $j_{n}$ and $h_{n}$ are the spherical Bessel and Hankel
functions of order $n$ respectively. This result can be derived using
the results in~\cite{vico_2014} for example.  From the asymptotic
expressions of spherical Bessel and Hankel functions, it follows that
$c_{n} = -1/4 + O(1/n)$ as $n\to \infty$.  Not handling the identity
term explicitly can affect the performance of quadrature methods (such
as Generalized Gaussian quadrature which is the method used in
\texttt{fmm3dbie}) as these methods rely on discretizing just the
principal value parts of operators.  No such issue arises when
$\Delta_{\Gamma}^{-1} q$ is replaced with $\mathcal S^2[\sigma]$.

\subsection{Harmonic fields}
Now we present the numerical results for the computation of the
harmonic vector fields on a twisted torus geometry whose boundary
$\Gamma$ is parametrized by $\bX: [0,2\pi]^2 \to \Gamma$ with
\begin{equation}
\bX(u,v) = \sum_{i=-1}^{2} \sum_{j=-1}^{2} \delta_{i,j}
\begin{pmatrix}
\cos{v} \cos{((1-i)u + jv)} \\
\sin{v} \cos{((1-i)u + jv)} \\
\sin{((1-i)u + jv)} \\
\end{pmatrix} \, ,
\end{equation}
where the non-zero coefficients are
\begin{equation}
  \begin{aligned}
    \delta_{-1,-1} &=0.17,  &\qquad \delta_{-1,0} &= 0.11,  &\qquad
    \delta_{0,0} &= 1,\\
   \delta_{1,0} &= 4.5, & \delta_{2,0} &= -0.25, & \delta_{0,1} &=
   0.01,\\
   & & \delta_{2,1} &= -0.45. & &
  \end{aligned}
\end{equation}
For this example, we define $\bV = \boldsymbol{\ell} \times (\bx -
\bx_{0})/| \bx -\bx_{0}|^{3}$, with $\bx_{0} = (0.2,0.2,0.2)$ and
$\boldsymbol{\ell} = (0,1,1)$ and then define $\bF = -\bn \times \bn
\times \bV$.  In~\cref{fig:htor}, we plot the tangential vector field
$\bF$, its curl-free component, its divergence-free component and the
harmonic vector field $\bH$. The second linearly independent harmonic
vector field can be obtained by computing $\bn \times \bH$. The
surface was discretized using $\Npat=600$ patches of order $p=9$.
In~\cref{tab:hodgestell}, we tabulate the results of a convergence
study. We estimating the error in the computation of the harmonic
vector fields by computing the $\mathbb{L}^{2}$ norms of
$\nabla_{\Gamma} \cdot \bH$ and $\nabla_{\Gamma} \cdot (\bn \times
\bH)$ using spectral differentiation. For this example $\| \bH \|
\approx 0.62498$, so the above error estimate serves as a proxy for a
relative error estimate as well. We observe the surface divergence and
surface curl converge to zero as indicated in~\cref{tab:hodgestell}.

\begin{table}[b!]
  \centering
\begin{tabular}{SSSSS} \toprule
    {$p$} & {$\Npat$} & {$N$} & {$||\nabla_{\Gamma}\cdot \mathbf{\bH}||_{\mathbb{L}^{2}(\Gamma)}$  } &  {$||\nabla_{\Gamma}\cdot (\mathbf{n} \times \mathbf{\bH})||_{\mathbb{L}^{2}(\Gamma)}$}  \\ \midrule
    9  & 150  & 6750 & 1.4E-1  & 1.1E-1    \\
    9  & 600  & 27000 & 1.4E-2  & 1.0E-2     \\
    9  & 2400  & 108000 & 3.2E-4 & 1.2E-4   \\
    9  & 9600  & 432000 & 1.8E-6 & 8.5E-7      \\\bottomrule
\end{tabular}
\caption{Error in the surface divergence of $\bH$ and $\bn \times \bH$
  for the computed harmonic vector field on the twisted
  torus. } \label{tab:hodgestell}
\end{table}

\begin{figure}[!t]
  \centering
\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{figures/stell_hodge_tanproj_20_4.png}
  \caption{Tangential vector field, $\mathbf{F}$}
  \label{fig:htor1}
\end{subfigure}%
\quad
\begin{subfigure}{0.45\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{figures/stell_hodge_cfree_20_4.png}
  \caption{Curl-free component, $ \nabla_{\Gamma}\alpha$}
  \label{fig:htor2}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{figures/stell_hodge_dfree_20_4.png}
  \caption{Div-free component, $\mathbf{n} \times \nabla_{\Gamma}\beta$  }
  \label{fig:htor3}
\end{subfigure}
\quad
\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=.8\linewidth]{figures/stell_hodge_harm_20_4.png}
  \caption{Harmonic component, $\mathbf{H}$}
  \label{fig:htor4}
\end{subfigure}
\caption{Hodge decomposition on a twisted torus with initial field
  $\bV$ given by a point current source at $\bx_{0}$.

  The color bar represents the magnitude of the vector field and the
  arrows represent the direction of the field.}
\label{fig:htor}
\end{figure}

We also compute the 20 linearly independent harmonic vector fields on
a genus 10 surface. For this geometry, we choose a random surface
vector field $\bV$ whose components are random tensor product Legendre
polynomials as discussed in~\cref{sec:harm-vec}. In~\cref{fig:g10}, we
plot the surface along with these 20 linearly independent harmonic
vector fields computed using our solver. The geometry was discretized
using $\Npat=2160$ patches of order $p=9$.  As before, we verify the
computations by calculating the $\mathbb{L}^{2}$ norms of the surface
divergence of $\bH$ and $\bn \times \bH$, where $\bH$ is the computed
harmonic field and observing that they converge to zero.
In~\cref{tab:hodgeg10}, we tabulate convergence results for a harmonic
vector field on the genus 10 surface computed using our solver.

\afterpage{
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{g10_gallery_crop.png}
    \caption{The 20 harmonic fields computed on a genus 10 geometry
      which was discretized with $\Npat=2160$ of order $p =9$.  The
      initial tangential vector field in each case is normalized to
      have $||\mathbf{F}||_{\mathbb{L}^{2}(\Gamma)} = 1$. The surface
      divergence of calculated harmonic field $\mathbf{H}$ in each
      case is $||\nabla_{\Gamma}\cdot
      \mathbf{H}||_{\mathbb{L}^{2}(\Gamma)} \approx O(10^{-1})$ and
      surface curl is $||\nabla_{\Gamma}\cdot \mathbf{n} \times
      \mathbf{H}||_{\mathbb{L}^{2}(\Gamma)} \approx O(10^{-1})$. The
      color bar represents the magnitude of the vector field and the
      arrows represent the direction of the field.}
    \label{fig:g10}
\end{figure}

\clearpage
}

\section{Conclusions and future work}
\label{sec:conclusions}

In this paper, we have presented a high-order FMM-accelerated
iterative solver for the numerical solution of the Laplace-Beltrami
equation on complex smooth surfaces in three dimensions. The
Laplace-Beltrami problem~$\Delta_\Gamma \psi = f$ is converted to a
second-kind integral equation by representing the solution as
$\psi=\mathcal S[\sigma]$ or $\psi=\mathcal S^2[\sigma]$, and using appropriate Cald\'eron
identities. The resulting integral equation, which requires the
application of various Laplace layer potentials, is then discretized
using a high-order method with locally corrected quadratures for
computing the weakly-singular layer potentials; their application is
accelerated using fast multipole methods.

While the integral equation can be written in several equivalent
analytic forms, we demonstrate the necessity of maintaining numerical
second-kindness by explicitly isolating the identity term of the
Fredholm equation in order to avoid stagnation of iterative solvers
such as GMRES. In some of these equivalent formulations, the
discretized operator
ends up having a high-dimensional spurious null space (proportional
to the number of patches used). This issue is related to the
difficulty of directly discretizing the differential form of the
Laplace-Beltrami equation when using patch-based discretizations of
surfaces and not enforcing smoothness across patches. However, both of
these issues can be mitigated by explicitly isolating the identity
part of the Fredholm equation.

\begin{table}[!t]
  \centering
  \begin{tabular}{SSSSS} \toprule
    {$p$} & {$\Npat$} & {$N$} & {$||\nabla_{\Gamma}\cdot \mathbf{H}||_{\mathbb{L}^{2}(\Gamma)}$  } &  {$||\nabla_{\Gamma}\cdot \mathbf{n} \times \mathbf{H}||_{\mathbb{L}^{2}(\Gamma)}$}  \\ \midrule
    9  & 2160  & 97200 & 5.6E-1  & 1.4E-1    \\
    9  & 8640  & 388800 & 6.1E-3 & 1.5E-3      \\
    9  & 34560  & 1555200 & 9.6E-5 & 4.5E-5      \\\bottomrule
\end{tabular}
\caption{Error in the surface divergence of $\bH$ and $\bn \times \bH$
  for one of the computed harmonic vector fields on the genus 10
  geometry. The computed harmonic vector field had $\mathbb{L}^{2}$
  norm of $0.66$} \label{tab:hodgeg10}
\end{table}

Finally, we also presented numerical examples demonstrating the
computation of harmonic vector fields on surfaces using the
Laplace-Beltrami solver. These vector fields are both of pure
mathematical interest and are also required for solving problems in
type I superconductivity and electromagnetism on topologically
non-trivial geometries.

There are still several open questions that remain to be
addressed. These include coupling high-order discretizations of the
Laplace-Beltrami integral equations to fast direct solvers;
subsequently using these solvers to develop iterative solvers for the
solution of Maxwell's equations using the generalized Debye
formulation; and extending these ideas for the solution of surface
diffusion problems that arise in pattern formation and cell
biology. These are all ongoing areas of research.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv