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# Tag Archives: BCS Theory

## Strontium Titanate – A Historical Tour

Like most ugly haircuts, materials tend to go in and out of style over time. Strontium titanate (SrTiO3), commonly referred to as STO, has, since its discovery, been somewhat timeless. And this is not just because it is often used as a substitute for diamonds. What I mean is that studying STO rarely seems to go out of style and the material always appears to have some surprises in store.

STO was first synthesized in the 1950s, before it was discovered naturally in Siberia. It didn’t take long for research on this material to take off. One of the first surprising results that STO had in store was that it became superconducting when reduced (electron-doped). This is not remarkable in and of itself, but this study and other follow-up ones showed that superconductivity can occur with a carrier density of only ~$5\times 10^{17} cm^{-3}$.

This is surprising in light of BCS theory, where the Fermi energy is assumed to be much greater than the Debye frequency — which is clearly not the case here. There have been claims in the literature suggesting that the superconductivity may be plasmon-induced, since the plasma frequency is in the phonon energy regime. L. Gorkov recently put a paper up on the arXiv discussing the mechanism problem in STO.

Soon after the initial work on superconductivity in doped STO, Shirane, Yamada and others began studying pure STO in light of the predicted “soft mode” theory of structural phase transitions put forth by W. Cochran and others. Because of an anti-ferroelectric structural phase transition at ~110K (depicted below), they we able to observe a corresponding soft phonon associated with this transition at the Brillouin zone boundary (shown below, taken from this paper). These results had vast implications for how we understand structural phase transitions today, when it is almost always assumed that a phonon softens at the transition temperature through a continuous structural phase transition.

Many materials similar to STO, such as BaTiO3 and PbTiO3, which also have a perovskite crystal structure motif, undergo a phase transition to a ferroelectric state at low (or not so low) temperatures. The transition to the ferroelectric state is accompanied by a diverging dielectric constant (and dielectric susceptibility) much in the way that the magnetic susceptibility diverges in the transition from a paramagnetic to a ferromagnetic state. In 1978, Muller (of Bednorz and Muller fame) and Burkard reported that at low temperature, the dielectric constant begins its ascent towards divergence, but then saturates at around 4K (the data is shown in the top panel below). Ferroelectricity is associated with a zone-center softening of a transverse phonon, and in the case of STO, this process begins, but doesn’t quite get there, as shown schematically in the image below (and you can see this in the data by Shirane and Yamada above as well).

Taken from Wikipedia

The saturation of the large dielectric constant and the not-quite-softening of the zone center phonon has led authors to refer to STO as a quantum paraelectric (i.e. because of the zero-point motion of the transverse optical zone-center phonon, the material doesn’t gain enough energy to undergo the ferroelectric transition). As recently as 2004, however, it was reported that one can induce ferroelectricity in STO films at room temperature by straining the film.

In recent times, STO has found itself as a common substrate material due to processes that can make it atomically flat. While this may not sound so exciting, this has had vast implications for the physics of thin films and interfaces. Firstly, this property has enabled researchers to grow high-quality thin films of cuprate superconductors using molecular beam epitaxy, which was a big challenge in the 1990’s. And even more recently, this has led to the discovery of a two-dimensional electron gas, superconductivity and ferromagnetism at the LAO/STO interface, a startling finding due to the fact that both materials are electrically insulating. Also alarmingly, when FeSe (a superconductor at around 7K) is grown as a monolayer film on STO, its transition temperature is boosted to around 100K (though the precise transition temperature in subsequent experiments is disputed but still high!). This has led to the idea that the FeSe somehow “borrows the pairing glue” from the underlying substrate.

STO is a gem of a material in many ways. I doubt that we are done with its surprises.

## Consistency in the Hierarchy

When writing on this blog, I try to share nuggets here and there of phenomena, experiments, sociological observations and other peoples’ opinions I find illuminating. Unfortunately, this format can leave readers wanting when it comes to some sort of coherent message. Precisely because of this, I would like to revisit a few blog posts I’ve written in the past and highlight the common vein running through them.

Condensed matter physicists of the last couple generations have grown up ingrained with the idea that “More is Different”, a concept first coherently put forth by P. W. Anderson and carried further by others. Most discussions of these ideas tend to concentrate on the notion that there is a hierarchy of disciplines where each discipline is not logically dependent on the one beneath it. For instance, in solid state physics, we do not need to start out at the level of quarks and build up from there to obtain many properties of matter. More profoundly, one can observe phenomena which distinctly arise in the context of condensed matter physics, such as superconductivity, the quantum Hall effect and ferromagnetism that one wouldn’t necessarily predict by just studying particle physics.

While I have no objection to these claims (and actually agree with them quite strongly), it seems to me that one rather (almost trivial) fact is infrequently mentioned when these concepts are discussed. That is the role of consistency.

While it is true that one does not necessarily require the lower level theory to describe the theories at the higher level, these theories do need to be consistent with each other. This is why, after the publication of BCS theory, there were a slew of theoretical papers that tried to come to terms with various aspects of the theory (such as the approximation of particle number non-conservation and features associated with gauge invariance (pdf!)).

This requirement of consistency is what makes concepts like the Bohr-van Leeuwen theorem and Gibbs paradox so important. They bridge two levels of the “More is Different” hierarchy, exposing inconsistencies between the higher level theory (classical mechanics) and the lower level (the micro realm).

In the case of the Bohr-van Leeuwen theorem, it shows that classical mechanics, when applied to the microscopic scale, is not consistent with the observation of ferromagnetism. In the Gibbs paradox case, classical mechanics, when not taking into consideration particle indistinguishability (a quantum mechanical concept), is inconsistent with the idea the entropy must remain the same when dividing a gas tank into two equal partitions.

Today, we have the issue that ideas from the micro realm (quantum mechanics) appear to be inconsistent with our ideas on the macroscopic scale. This is why matter interference experiments are still carried out in the present time. It is imperative to know why it is possible for a C60 molecule (or a 10,000 amu molecule) to be described with a single wavefunction in a Schrodinger-like scheme, whereas this seems implausible for, say, a cat. There does again appear to be some inconsistency here, though there are some (but no consensus) frameworks, like decoherence, to get around this. I also can’t help but mention that non-locality, à la Bell, also seems totally at odds with one’s intuition on the macro-scale.

What I want to stress is that the inconsistency theorems (or paradoxes) contained seeds of some of the most important theoretical advances in physics. This is itself not a radical concept, but it often gets neglected when a generation grows up with a deep-rooted “More is Different” scientific outlook. We sometimes forget to look for concepts that bridge disparate levels of the hierarchy and subsequently look for inconsistencies between them.

## Paradigm Shifts and “The Scourge of Bibliometrics”

Yesterday, I attended an insightful talk by A.J. Leggett at the APS March Meeting entitled Reflection on the Past, Present and Future of Condensed Matter Physics. The talk was interesting in two regards. Firstly, he referred to specific points in the history of condensed matter physics that resulted in (Kuhn-type) paradigm shifts in our thinking of condensed matter. Of course these paradigm shifts were not as violent as special relativity or quantum mechanics, so he deemed them “velvet” paradigm shifts.

This list, which he acknowledged was personal, consisted of:

1. Landau’s theory of the Fermi liquid
2. BCS theory
3. Renormalization group
4. Fractional quantum hall effect

Notable absentees from this list were superfluidity in 3He, the integer quanutm hall effect, the discovery of cuprate superconductivity and topological insulators. He argued that these latter advances did not result in major conceptual upheavals.

He went on to elaborate the reasons for these velvet revolutions, which I enumerate to correspond to the list above:

1. Abandonment of microscopic theory, in particular with the use of Landau parameters; trying to relate experimental properties to one another with the input of experiment
2. Use of an effective low-energy Hamiltonian to describe phase of matter
3. Concept of universality and scaling
4. Discovery of quasiparticles with fractional charge

It is informative to think about condensed matter physics in this way, as it demonstrates the conceptual advances that we almost take for granted in today’s work.

The second aspect of his talk that resonated strongly with the audience was what he dubbed “the scourge of bibliometrics”. He told the tale of his own formative years as a physicist. He published one single-page paper for his PhD work. Furthermore, once appointed as a lecturer at the University of Sussex, his job was to be a lecturer and teach from Monday thru Friday. If he did this job well, it was considered a job well-done. If research was something he wanted to partake in as a side-project, he was encouraged to do so. He discussed how this atmosphere allowed him to develop as a physicist, without the requirement of publishing papers for career advancement.

Furthermore, he claimed, because of the current focus on metrics, burgeoning young scientists are now encouraged to seek out problems that they can solve in a time frame of two to three years. He saw this as a terrible trend. While it is often necessary to complete short-term projects, it is also important to think about problems that one may be able to solve in, say, twenty years, or maybe even never. He claimed that this is what is meant by doing real science — jumping into the unknown. In fact, he asserted that if he were to give any advice to graduate students, postdocs and young faculty in the audience, it would be to try to spend about 20% of one’s time committed to some of these long-term problems.

This raises a number of questions in my mind. It is well-acknowledged within the community and even the blogosphere that the focus on publication number and short term-ism within the condensed matter physics community is detrimental. Both Ross McKenzie and Doug Natelson have expressed such sentiment numerous times on their blogs as well. From speaking to almost every physicist I know, this is a consensus opinion. The natural question to ask then is: if this is the consensus opinion, why is the modern climate as such?

It seems to me like part of this comes from the competition for funding among different research groups and funding agencies needing a way to discriminate between them. This leads to the widespread use of metrics, such as h-indices and publication number, to decide whether or not to allocate funding to a particular group. This doesn’t seem to be the only reason, however. Increasingly, young scientists are judged for hire by their publication output and the journals in which they publish.

Luckily, the situation is not all bad. Because so many people openly discuss this issue, I have noticed that the there is a certain amount of push-back from individual scientists. On my recent postdoc interviews, the principal investigators were most interested in what I was going to bring to the table rather than peruse through my publication list. I appreciated this immensely, as I had spent a large part of my graduate years pursuing instrumentation development. Nonetheless, I still felt a great deal of pressure to publish papers towards the end of graduate school, and it is this feeling of pressure that needs to be alleviated.

Strangely, I often find myself in the situation working despite the forces that be, rather than being encouraged to do so. I highly doubt that I am the only one with this feeling.

## Reflecting on General Ideas

In condensed matter physics, it is easy to get lost in the details of one’s day-to-day work. It is important to sometimes take the time to reflect upon what you’ve done and learned and think about what it all means. In this spirit, below is a list of some of the most important ideas related to condensed matter physics that I picked up during my time as an undergraduate and graduate student. This is of course personal, and I hope that in time I will add to the list.

1. Relationship between measurements and correlation functions
2. Relationship between equilibrium fluctuations and non-equilibrium dissipative channels (i.e. the fluctuation-dissipation theorem)
3. Principle of entropy maximization/free-energy minimization for matter in equilibrium
4. Concept of the quasi-particle and screening
5. Concept of Berry phase and the corresponding topological and geometrical consequences
6. Broken symmetry, the Landau paradigm of phase classification and the idea of an order parameter
7. Sum rules and the corresponding constraints placed on both microscopic theories and experimental spectra
8. Bose-Einstein and Cooper Pair condensation and their spectacular properties
9. Logical independence of physical theories on the theory of everything
10. Effects of long-range vs. short-range interactions on macroscopic properties of solids
11. Role of dimensionality in observing qualitatively different physical properties and phases of matter

The first two items on the list are well-explained in Forster’s Hydrodynamics, Fluctuations, Broken Symmetry and Correlation Functions without the use of Green’s functions and other advanced theoretical techniques. Although not yet a condensed matter phenomenon, Bell’s theorem and non-locality rank among the most startling consequences of quantum mechanics that I learned in graduate school. I suspect that its influence will be observed in a condensed matter setting in due time.

## What is Scientific Consensus?

When a theory is put forward, it takes time for the scientific community to evaluate its merits. Ultimately, one hopes that the theory is able to not only explain past data, but to be able to predict the outcome of future experiments as well. When the dust settles, we hope that we reach “scientific consensus” regarding a theory. But what does this mean?

Since this is a condensed matter blog, let us take BCS theory as an example. When BCS was formulated, it was able to explain numerous experimental observations, such as the evolution of the electronic gap as a function of temperature as well as the specific heat anomaly among several other observations. However, there were also apparent problems with BCS theory. Many physicists were concerned with the non-conservation of particle number and with some aspects of broken gauge symmetry (pdf!) in the theory. Notably also, there were materials that did not conform exactly to the BCS formulas, such as Pb (lead), where the predicted $2\Delta/k_BT_c=3.5$ relation and was found instead to be around 4.38.

So the question is, how were these issues resolved and how did the community reach the general consensus that BCS theory was applicable for the existing superconductors at that point in history?

This question actually leads to a more general scientific question: how do we reach a consensus concerning a theory? The answer to this question involves a Bayesian approach. We start with a prior probability based on our biases and update this prior probability as we begin to examine more and more data, making predictions as we go along. If physicist A had spent the past 10 years working actively on a theory of superconductivity and may secretly hope that BCS theory is wrong, s/he may start out with only 3% confidence that BCS theory is correct. On the other hand, physicist B may be completely neutral and would have a prior probability of 50%. Another physicist C would perhaps be swayed by the fact that Bardeen had just won a Nobel prize in physics for the invention of the transistor and therefore has a initial confidence level of 85% that BCS is correct. These constitute these physicists’ prior probabilities or “biases”.

What happens with time? Well, BCS predicted the existence of the Hebel-Slichter peak in the NMR spectrum, which was then observed shortly thereafter. Furthermore, Anderson showed that one could project out a particle-conserving part of the ground-state, which resolved some theoretical issues pertaining to particle-number conservation. Gorkov was also able to show that the phenomenological equations of Ginzburg and Landau were derivable from BCS theory (pdf!). McMillan and Rowell then conducted their famous experiments where they analyzed the second derivative of tunneling spectra, which exhibited phonon anomalies, to explain why lead did not obey the simple BCS formalism, but required a small extension.

As these data points accumulated, confidence in BCS theory grew for physicists A, B and C. In a Bayesian picture, we update our beliefs as we get more and more data points that are consistent with (or resolve questions pertaining to) a particular theory. Ultimately, the members of the scientific community would asymptotically approach a place where they understand the domain of validity of BCS theory and understand what it can predict. The picture I have in mind to represent this process is plotted below:

This plot is of a Bayesian updating scheme based on prior beliefs. The convergence of the viewpoints of physicists A, B and C is what is crudely meant by scientific consensus. Note that a person that starts out with a dogmatic 0% belief in the correctness of BCS theory will not change his/her mind with time.

It is important to emphasize that what I have called the 100% confidence level in my plot is meant to indicate a place where we understand the limitations and validity of a theory and how/when to apply this theory. For example, we can have 99.9% confidence that Newton’s theory of gravity will enable us to solve simple kinematics problems on the surface of the earth. While we know that Newton’s theory of gravity requires corrections from Einstein’s theory of general relativity, our confidence in Newton’s theory is not diminished when used in the correct limits. Therefore, in this Bayesian scheme, we get closer and closer to being 100% confident in a theory, but never quite reach it.

This is a rather Popperian view of scientific consensus and we know the limits of such a view in light of Kuhn’s work, but I think it nonetheless serves as a valuable guide as to how to think about the concept which is so often corrupted, especially in regard to the climate change discussion. Therefore, in the future, when people talk about scientific consensus, think convergence and think Bayes.

## Why Was BCS So Important?

BCS theory, which provides a microscopic framework to understand superconductivity, made us realize that a phenomenon similar to Bose-Einstein condensation was possible for fermions. This is far from a trivial statement, though we sometimes think of it as so in present times.

A cartoon-y scheme to understand it is in the following way. We know that if you put a few fermions together, you can get a boson, such as 4-helium. It was also known well before BCS theory, that one gets a phenomenon reminiscent of Bose-Einstein condensation, known as superfluidity, in 4-helium below 2.17K. The view of 4-helium as a Bose-Einstein condensate (BEC) was advocated strongly by Fritz London, who was perhaps the first to think of it in such a way.

Now let us think of another type of boson, a diatomic molecule, as seen in gas form below:

Even if the individual atoms were fermions, one would then predict that if this bosonic diatomic gas molecule could remain in the gas phase all the way down to low temperature, that at some point, this diatomic gas would condense into a BEC. This idea is correct and this is indeed what is observed.

However, the idea of a BEC becomes a little more cloudy when one considers a less dilute diatomic gas where the atoms are not so strongly bound together. In that case, the cartoon starts to look something like this:
Here the “diatomic molecules” are overlapping, and it is not easy to see which atoms are paired together to form the diatomic molecule, if one can even ascribe this trait to them. In this case, it is no longer simple to see whether or not BEC will occur and indeed if there is a limit in distance between the molecules that will necessarily give rise to BEC.

This is the question that BCS theory so profoundly addresses. It says that the “diatomic molecules” or Cooper pairs can span a great distance. In superconducting aluminum, this distance is ~16,000 Angstroms, which means the Cooper pairs are wildly overlapping. In fact, in this limit, the Cooper pair is no longer strictly even a boson, in the sense that Cooper pair creation and annihilation operators do not obey Bose-Einstein commutation relations.

However, the Cooper pair can still qualitatively thought of as a pseudo-boson that undergoes pseud0-BEC, and this picture is indeed  very useful. It enabled the prediction of pseud0-BEC in neutron stars, liquid 3-helium and ultra-cold fermionic gases — predictions which now have firm experimental backing.

An interesting note is that one can study this BCS-to-BEC crossover in ultracold Fermi gases and go from the overlapping to non-overlapping limit by tuning the interaction between atoms and I’ll try to write a post about this in the near future.

So while BCS theory has many attributes that make it important, to my mind, the most profound thing is that it presents a mechanism by which weakly interacting fermionic pairs can condense into a pseudo-BEC. This is not at all obvious, but indeed what happens in nature.

Update: In light of the description above, it seems surprising that the temperature at which Cooper pairs form is the same temperature at which they seem to condense into a pseudo-BEC. Why this is the case is not obvious and I think is an open question, especially with regards to the cuprates and in particular the pseudogap.

## A Critical Ingredient for Cooper Pairing in Superconductors within the BCS Picture

I’m sure that readers that are experts in superconductivity are aware of this fact already, but there is a point that I feel is not stressed enough by textbooks on superconductivity. This is the issue of reduced dimensionality in BCS theory. In a previous post, I’ve shown the usefulness of the thinking about the Cooper problem instead of the full-blown BCS solution, so I’ll take this approach here as well. In the Cooper problem, one assumes a 3-dimensional spherical Fermi surface like so:

3D Fermi Surface

What subtly happens when one solves the Cooper problem, however, is the reduction from three dimensions to two dimensions. Because only the electrons near the Fermi surface condense, one is really working in a shell around the Fermi surface like so, where the black portion does not participate in the formation of Cooper pairs:

Effective 2D Topology Associated with the Cooper Problem

Therefore, when solving the Cooper problem, one goes from working in a 3D solid sphere (the entire Fermi sea), to working on the surface of the sphere, effectively a 2D manifold. Because one is now confined to just the surface, it enables one of the most crucial steps in the Cooper problem: assuming that the density of states ($N(E)$) at the Fermi energy is a constant so that one can pull it out of the integral (see, for example, equation 9 in this set of lecture notes by P. Hirschfeld).

The more important role of dimensionality, though, is in the bound state solution. If one solves the Schrodinger equation for the delta-function potential (i.e. $V(x,y)= -\alpha\delta(x)\delta(y)$) in 2D one sees a quite stunning (but expected) resemblance to the Cooper problem. It tuns out that the solution to obtain a bound state takes the following form:

$E \sim \exp{(-\textrm{const.}/\alpha)}$.

Note that this is exactly the same function that appears in the solution to the Cooper problem, and this is of course not a coincidence. This function is not expandable in terms of a Taylor series, as is so often stressed when solving the Cooper problem and is therefore not amenable to perturbation methods. Note, also, that there is a bound state solution to this problem whenever $\alpha$ is finite, again similar to the case of the Cooper problem. That there exists a bound state solution for any $\alpha >0$ no matter how small, is only true in dimensions two or less. This is why reduced dimensionality is so critical to the Cooper problem.

Furthermore, it is well-known to solid-state physicists that for a Fermi gas/liquid, in 3D $N(E) \sim \sqrt{E}$, in 2D $N(E) \sim$const., while in 1D $N(E) \sim 1/\sqrt{E}$. Hence, if one is confined to two-dimensions in the Cooper problem, one is able to treat the density of states as a constant, and pull this term out of the integral (see equation 9 here again) even if the states are not confined to the Fermi energy.

This of course raises the question of what happens in an actual 2D or quasi-2D solid. Naively, it seems like in 2D, a solid should be more susceptible to the formation of Cooper pairs including all the electrons in the Fermi sea, as opposed to the ones constrained to be close to the Fermi surface.

If any readers have any more insight to share with respect to the role of dimensionality in superconductivity, please feel free to comment below.


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# If 2x/ (root(-3x^2 + 27)) is not a real number

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10 Jul 2018, 00:48
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If $$\frac{2x}{\sqrt{-3x^2 + 27}}$$ is not a real number, which of the following specifies all the possible values of x?

A. $$x\leq-3$$ or $$x\geq3$$
B. $$x\leq-3$$
C. $$x\geq3$$
D. $$x\leq-4$$ or $$x\geq4$$
E. $$-3 \leq x \leq 3$$
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If 2x/ (root(-3x^2 + 27)) is not a real number  [#permalink]

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10 Jul 2018, 00:55
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GIven $$\frac{2x}{\sqrt{-3x^2 + 27}}$$ is not a real number

=> The number inside the square root sign is < 0. Also since the square root is in denominator, the number is not a real number if denominator is 0

=> $$-3x^2 + 27 \leq 0$$

=> $$3x^2 - 27 \geq 0$$

=> $$3x^2 \geq 27$$

=> $$x^2 \geq 9$$

=> $$x\leq-3$$ or $$x\geq3$$

Hence option A
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Re: If 2x/ (root(-3x^2 + 27)) is not a real number  [#permalink]

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10 Jul 2018, 03:54
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Karthik200 wrote:
If $$\frac{2x}{\sqrt{-3x^2 + 27}}$$ is not a real number, which of the following specifies all the possible values of x?

A. $$x\leq-3$$ or $$x\geq3$$
B. $$x\leq-3$$
C. $$x\geq3$$
D. $$x\leq-4$$ or $$x\geq4$$
E. $$-3 \leq x \leq 3$$

Since the GMAT is constrained to real numbers, I would disregard this problem.
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Re: If 2x/ (root(-3x^2 + 27)) is not a real number  [#permalink]

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10 Jul 2018, 04:03
GMATGuruNY wrote:

Since the GMAT is constrained to real numbers, I would disregard this problem.

Good to know about this constraint. Thank you.
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Re: If 2x/ (root(-3x^2 + 27)) is not a real number  [#permalink]

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15 Jul 2018, 09:13
Although I got this question right, I am just curious to check is this a GMAT like question? I was told by someone that imaginary numbers are not tasted on GMAT.
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Re: If 2x/ (root(-3x^2 + 27)) is not a real number  [#permalink]

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18 Jul 2018, 17:22
workout wrote:
GIven $$\frac{2x}{\sqrt{-3x^2 + 27}}$$ is not a real number

=> The number inside the square root sign is < 0. Also since the square root is in denominator, the number is not a real number if denominator is 0

=> $$-3x^2 + 27 \leq 0$$

=> $$3x^2 - 27 \geq 0$$

=> $$3x^2 \geq 27$$

=> $$x^2 \geq 9$$

=> $$x\leq-3$$ or $$x\geq3$$

Hence option A

Thanks workout for the nice explanation
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Re: If 2x/ (root(-3x^2 + 27)) is not a real number  [#permalink]

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18 Jul 2018, 18:56
-3x^2+27<0
3x^2-27>0
3(x^2-9)>0
x^2-9>0
(x+3)(x-3)>0

x+3>0 and x-3>0
x>-3 and x>3

x+3<0 and x-3<0
x<-3 and x<3
Answer A

Grade school algebra questions rly comfortable easy
Re: If 2x/ (root(-3x^2 + 27)) is not a real number &nbs [#permalink] 18 Jul 2018, 18:56
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# If 2x/ (root(-3x^2 + 27)) is not a real number

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# isBounded -- determines if a poset is bounded

## Synopsis

• Usage:
i = isBounded P
• Inputs:
• P, an instance of the type Poset,
• Outputs:
• i, , whether $P$ is bounded

## Description

The poset $P$ is bounded if it has a unique minimal element and a unique maximal element.

The $n$ chain and $n$ booleanLattice are bounded.

 i1 : n = 5; i2 : isBounded chain n o2 = true i3 : B = booleanLattice n; i4 : isBounded B o4 = true

The middle ranks of an $n$ boolean lattice are not bounded.

 i5 : isBounded flagPoset(B, {1,2,3,4}) o5 = false


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Authors Title Attachments
C Bandle; MdM Gonzalez; MA Fontelos; N Wolanski A nonlocal diffusion problem on manifolds
D Bucur; A Henrot A new isoperimetric inequality for the elasticae
GG Chen; M Feldman Free boundary problems in shock reflection/diffraction and related transonic flow problems
D Bucur; D Mazzoleni A surgery result for the spectrum of the Dirichlet Laplacian
L Caffarelli; JL Vázquez Regularity of solutions of the fractional porous medium flow with exponent 1/2
K Yeressian Obstacle problem with a degenerative force term
GG Chen; M Perepelitsa Vanishing viscosity solutions of the compressible Euler equations with spherical symmetry and large initial data
CM Elliott; T Ranner A computational approach to an optimal partition problem on surfaces
CM Elliott; H Fritz Time-periodic solutions of advection-diffusion equations on moving hypersurfaces
G Canevari; A Segatti; M Veneroni Morse's index formula in VMO for compact manifolds with boundary
A Constantin; W Strauss; E Varvaruca Global bifurcation of steady gravity water waves with critical layers
M Malin; VD Radulescu Competition phenomena for difference equations with oscillatory nonlinearities
K Yeressian On generalized solutions of axisymmetric two-phase incompressible viscous flow with surface tension
C Bandle; A Wagner Two Robin boundary value problems with opposite sign
K Yeressian Nondegeneracy in the obstacle problem with a degenerate force term
B Alleche; VD Radulescu; M Sebaoui The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities
M Hintermüller; D Wegner Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies
B Kawohl; C Nitsch; G Sweers More on the potential for the farthest-point distance function
P Colli; G Gilardi; D Hilhorst On a Cahn-Hilliard type phase field system related to tumor growth
A Segatti; M Snarski; M Veneroni Analysis of a variational model for nematic shells
S Bartels A simple scheme for the approximation of elastic vibrations of inextensible curves
CB Davis; SW Walker The saddle-point formulation and finite element method for the Stefan problem with surface tension
A Alphonse; CM Elliott; B Stinner An abstract framework for parabolic PDEs on evolving spaces
CM Elliott; C Venkataraman Error analysis for an ALE evolving surface finite element method
JF Rodrigues On the mathematical analysis of thick fluids
JL Vázquez The mesa problem for the fractional porous medium equation
C Bandle; A Wagner Isoperimetric inequalities for the principal eigenvalue of a membrane and the energy of problems with Robin boundary conditions
S Bartels Fast and accurate finite element approximation of wave maps into spheres
D Stan; F del Teso; JL Vázquez Transformations of self-similar solutions for porous medium equations of fractional type
H Mikayelyan; H Shaghholian Hopf's lemma for a class of singular/degenerate PDE-s
D E Apushkinskaya; N N Uraltseva On regularity properties of solutions to hysteresis-type problems
G Buttazzo; M van den Berg; B Velichkov Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity
M Chipot; T Savitska Nonlocal $\it p$-laplace equations depending on the $\it L^p$ norm of the gradient
S Challal; A Lyaghfouri; JF Rodrigues; R Teymurazyan On the regularity of the free boundary for quasilinear obstacle problems
B Alleche; VD Radulescu Equilibrium problems techniques in the qualitative analysis of quasi-hemivariational inequalities
A Bahrouni; H Ounaies; VD Radulescu Infinitely many solutions for a class of sublinear Schrödinger equations with indefinite potentials


text:
# Clyde is a cash-method taxpayer who reports on a calendar-year basis

 Clyde is a cash-method taxpayer who reports on a calendar-year basis. This year Paylate Corporation has decided to pay Clyde a year-end bonus of $1,000.  Determine the amount Clyde should include in his gross income this year under the following circumstances: a. Paylate Corporation wrote the check and put it in his office mail slot on December 30 of this year, but Clyde did not bother to stop by the office to pick it up until after year-end. Amount to be included: b. Paylate Corporation mistakenly wrote the check for$100. Clyde received the remaining \$900 after year-end.

Amount to be included:

c.

Paylate Corporation mailed the check to Clyde before the end of the year, (and it was delivered before year end). Although Clyde expected the bonus payment, he decided not to collect his mail until after year-end.

Amount to be included:

 d. Clyde picked up the check in December, but the check could not be cashed immediately because it was postdated January 10. Amount to be included:


text:
# An hp-Hybrid High-Order Method for Variable Diffusion on General Meshes

Joubine Aghili, Daniele A. Di Pietro and Berardo Ruffini

# Abstract

In this work, we introduce and analyze an hp-hybrid high-order (hp-HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We prove hp-convergence estimates for both the energy- and L2-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novel hp-approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.

MSC 2010: 65N30; 65N08; 76R50

## 1 Introduction

Over the last few years, discretization technologies have appeared that support arbitrary approximation orders on general polytopal meshes. In this work, we focus on a particular instance of such technologies, the so-called Hybrid High-Order (HHO) methods originally introduced in [18, 17]; see also [21] for an introduction. So far, the literature on HHO methods has focused on the h-version of the method with uniform polynomial degree. Our goal is to provide a first example of variable-degree hp-HHO method and carry out a full hp-convergence analysis valid for fairly general meshes and arbitrary space dimension. Let Ωd, d1, denote a bounded connected polytopal domain with Lipschitz boundary. We consider the variable diffusion model problem: Find u:Ω such that

(1.1){-(𝜿u)=fin Ω,u=0on Ω,

where 𝜿 is a uniformly positive, symmetric, tensor-valued field on Ω, while fL2(Ω) denotes a volumetric source. For the sake of simplicity, we assume that 𝜿 is piecewise constant on a partition PΩ of Ω into polytopes. The weak formulation of problem (1.1) reads: Find uU:=H01(Ω) such that

(1.2)(𝜿u,v)=(f,v)for all vU,

where we have used the notation (,) for the usual inner products of both L2(Ω) and L2(Ω)d. Here, the scalar-valued field u represents a potential, and the vector-valued field 𝜿u the corresponding flux.

For a given polytopal mesh 𝒯h={T} of Ω, the hp-HHO discretization of problem (1.2) is based on two sets of degrees of freedom (DOFs):

1. skeletal DOFs, consisting in (d-1)-variate polynomials of total degree pF0 on each mesh face F, and

2. elemental DOFs, consisting in d-variate polynomials of degree pT on each mesh element T, where pT denotes the lowest degree of skeletal DOFs on the boundary of T.

Skeletal DOFs are globally coupled, and can be alternatively interpreted as traces of the potential on the mesh faces or as Lagrange multipliers enforcing the continuity of the normal flux at the discrete level; cf. [1, 12] for further insight. Elemental DOFs, on the other hand, are auxiliary DOFs that can be locally eliminated by static condensation, as detailed in [12, Section 2.4] for the case where pF=p for all mesh faces F.

Two key ingredients are devised locally from skeletal and elemental DOFs attached to each mesh element T:

1. a reconstruction of the potential of degree (pT+1) (i.e., one degree higher than elemental DOFs in T) obtained solving a small Neumann problem, and

2. a stabilization term penalizing face-based residuals and polynomially consistent up to degree (pT+1).

The local contributions obtained from these two ingredients are then assembled following a standard, finite element-like procedure. The resulting discretization has several appealing features, the most prominent of which are summarized hereafter:

1. It is valid for fairly general polytopal meshes.

2. The construction is dimension-independent, which can significantly ease the practical implementation.

3. It enables the local adaptation of the approximation order, a highly desirable feature when combined with a regularity estimator (whose development will be addressed in a separate work).

4. It exhibits only a moderate dependence on the diffusion coefficient 𝜿.

5. It has a moderate computational cost thanks to the possibility of eliminating elemental DOFs locally via static condensation.

6. Parallel implementations can be simplified by the fact that processes communicate via skeletal unknowns only.

The seminal works on the p- and hp-conforming finite element method on standard meshes date back to the early 80s; cf. [7, 6, 5]. Starting from the late 90s, nonconforming methods on standard meshes supporting arbitrary-order have received a fair amount of attention; a (by far) nonexhaustive list of contributions focusing on scalar diffusive problems similar to the one considered here includes [29, 10, 28, 24, 31]. The possibility of refining both in h and in p on general meshes is, on the other hand, a much more recent research topic. We cite, in particular, hp-composite [3, 25] and polyhedral [9] discontinuous Galerkin methods, and the two-dimensional virtual element method proposed in [8].

The main novelty of the paper are hp-energy- and L2-estimates for HHO methods summarized in Section 3.2. These are the first results of this kind for HHO methods, and among the first for discontinuous skeletal methods in general (a prominent example of discontinuous skeletal methods are the hybridizable discontinuous Galerkin methods of [13]; cf. [12] for a precise study of their relation with HHO methods).

The cornerstone of the analysis is a novel extension of the classical Babuška–Suri hp-approximation results to regular mesh sequences in the sense of [16, Chapter 1] and arbitrary space dimension d1; see Lemma 2.3 below. Similar results had been derived in [8] for d=2 and, under assumptions on the mesh different from the ones proposed here, in [9] for d{2,3}. A relevant difference with respect to classical approximation results based on the Bramble–Hilbert lemma (see, e.g., [23, Theorem 1.103]) is that we cannot rely here on the notion of reference element, since general polyhedral shapes are allowed. As a result, it turns out that a key point consists in showing that the regularity assumptions adapted to polyhedral meshes imply uniform bounds for the Lipschitz constant of mesh elements.

The resulting energy-norm estimate confirms the characteristic h-superconvergence behavior of HHO methods, whereas we have a more standard scaling as (pT+1)-pT with respect to the polynomial degree pT of elemental DOFs. This scaling is analogous to the best available results for discontinuous Galerkin (dG) methods on rectangular meshes based on polynomials of degree pT, cf. [24] (on more general meshes, the scaling for the symmetric interior penalty dG method is pT-(pT-1/2), half a power less than for the hp-HHO method studied here). Classically, when elliptic regularity holds, the h-convergence order can be increased by 1 for the L2-norm. In our error estimates, the dependence on the diffusion coefficient is carefully tracked, showing full robustness with respect to its heterogeneity and only a moderate dependence with a power of 12 on its local anisotropy when the error in the energy-norm is considered. These results improve the ones in [19] where, considering a different norm for the error, a power of 1 was found. Numerical experiments confirm the expected exponentially convergent behavior for both isotropic and strongly anisotropic diffusion coefficients on a variety of two-dimensional meshes.

The rest of the paper is organized as follows. In Section 2 we introduce the notation and state the basic results required in the analysis including, in particular, Lemma 2.3 (whose proof is detailed in Appendix A). In Section 3 we formulate the hp-HHO method, state our main results, and provide some numerical examples. The proofs of the main results, preceded by the required preparatory material, are collected in Section 4.

## 2 Setting

In this section we introduce the notation and state the basic results required in the analysis.

### 2.1 Mesh and Notation

Let *+ denote a countable set of meshsizes having 0 as its unique accumulation point. We consider mesh sequences (𝒯h)h where, for all h, 𝒯h={T} is a finite collection of nonempty disjoint open polytopal elements such that Ω¯=T𝒯hT¯ and h=maxT𝒯hhT (hT stands for the diameter of T). A hyperplanar closed connected subset F of Ω¯ is called a face if it has positive (d-1)-dimensional measure and

1. either there exist distinct T1,T2𝒯h such that F=T1T2 (and F is an interface)

2. or there exists T𝒯h such that F=TΩ (and F is a boundary face).

Interfaces are collected in hi, boundary faces in hb, and we let h:=hihb. It is assumed that the set of faces h partitions the mesh skeleton in the sense that distinct faces have disjoint interiors and that T𝒯hT=FhF. For all T𝒯h, the set T:={FhFT} collects the faces lying on the boundary of T and, for all FT, we denote by 𝒏TF the normal vector to F pointing out of T.

The following assumptions on the mesh will be kept throughout the exposition.

### Assumption 2.1 (Admissible Mesh Sequence).

We assume that (𝒯h)h is admissible in the sense of [16, Chapter 1], i.e., for all h, 𝒯h admits a matching simplicial submesh 𝔗h and there exists a real number ϱ>0 (the mesh regularity parameter) independent of h such that the following conditions hold:

1. ϱhSrS for all h and all simplices S𝔗h of diameter hS and inradius rS.

2. ϱhThS for all h, all T𝒯h, and all S𝔗h such that ST.

3. Every mesh element T𝒯h is star-shaped with respect to every point of a ball of radius ϱhT.

### Assumption 2.2 (Compliant Mesh Sequence).

We assume that the mesh sequence is compliant with the partition PΩ on which the diffusion tensor 𝜿 is piecewise constant, so that jumps only occur at interfaces and, for all T𝒯h, we set

𝜿T:=𝜿|T0(T)d×d.

In what follows, for all T𝒯h, κ¯T and κ¯T denote the largest and smallest eigenvalue of 𝜿T, respectively, and λ𝜿,T:=κ¯T/κ¯T the local anisotropy ratio.

### 2.2 Basic Results

For a subset X of Ω and an index q, Hq(X) denotes the Hilbert space of functions which are in L2(X) together with their weak derivatives of order q, equipped with the usual inner product (,)q,X and associated norm q,X. When q=0, we recover the Lebesgue space L2(X), and the subscript 0 is omitted from both the inner product and the norm. The subscript X is also omitted when X=Ω. For a given integer l0, we denote by l(X) the space spanned by the restriction to X of d-variate polynomials of degree l. For further use, we also introduce the L2-projector πXl:L1(X)l(X) such that, for all wL1(X),

(2.1)(πXlw-w,v)X=0for all vl(X).

We recall hereafter a few known results on admissible mesh sequences and refer to [16, Chapter 1] and [14, 15] for a more comprehensive collection. By [16, Lemma 1.41], there exists an integer N(d+1) (possibly depending on d and ϱ) such that the maximum number of faces of one mesh element is bounded,

(2.2)maxh,T𝒯hcard(T)N.

The following multiplicative trace inequality, valid for all h, all T𝒯h, and all vH1(T), is proved in [16, Lemma 1.49]:

(2.3)vT2C(vTvT+hT-1vT2),

where C only depends on d and ϱ. We also note the following local Poincaré’s inequality valid for all T𝒯h and all vH1(T) such that (v,1)T=0:

(2.4)vTCPhTvT,

where CP=π-1 when T is convex, while it can be estimated in terms of ϱ for nonconvex elements (cf., e.g., [34]).

The following functional analysis results lie at the heart of the hp-analysis carried out in Section 4.

### Lemma 2.3 (Approximation).

There is a real number C>0 (possibly depending on d and ϱ) such that, for all hH, all TTh, all integers l1, all s0, and all vHs+1(T), there exists a polynomial ΠTlvPl(T) satisfying, for all 0qs+1,

(2.5)v-ΠTlvq,TChTmin(l,s)-q+1ls+1-qvs+1,T.

### Proof.

See Appendix A. ∎

### Lemma 2.4 (Discrete Trace Inequality).

There is a real number C>0 (possibly depending on d and ϱ) such that, for all hH, all TTh, all integers l1, and all vPl(T), it holds

(2.6)vTClhT1/2vT.

### Proof.

When all meshes in the sequence (𝒯h)h are simplicial and conforming, the proof of (2.6) can be found in [30, Theorem 4.76] for d=2; for d2 the proof is analogous. The extension to admissible mesh sequences in the sense of Assumption 2.1 can be done following the reasoning in [16, Lemma 1.46]. ∎

## 3 Discretization

In this section, we formulate the hp-HHO method, state our main results, and provide some numerical examples.

### 3.1 The hp-HHO Method

We present in this section an extension of the classical HHO method of [18] accounting for variable polynomial degrees. Let a vector p¯h=(pF)Fhh of skeletal polynomial degrees be given. For all T𝒯h, we denote by p¯T=(pF)FT the restriction of p¯h to T, and we introduce the following local space of DOFs:

(3.1)U¯Tp¯T:=pT(T)×(×FTpF(F)),pT:=minFTpF.

The standard HHO notation v¯T=(vT,(vF)FT) is used for a generic element of U¯Tp¯T. We define the local potential reconstruction operator

rTpT+1:U¯Tp¯TpT+1(T)

such that, for all v¯TU¯Tp¯T and all wpT+1(T),

(3.2)(𝜿TrTpT+1v¯T,w)T=-(vT,(𝜿Tw))T+FT(vF,𝜿Tw𝒏TF)F

and

(rTpT+1v¯T-vT,1)T=0.

Note that computing rTpT+1v¯T according to (3.2) requires to invert the 𝜿T-weighted stiffness matrix of k+1(T), which can be efficiently accomplished by a Cholesky solver.

We define on U¯Tp¯T×U¯Tp¯T the local bilinear form aT such that

(3.3)aT(u¯T,v¯T):=(𝜿TrTpT+1u¯T,rTpT+1v¯T)T+sT(u¯T,v¯T),
(3.4)sT(u¯T,v¯T):=FTκFhT(δTFp¯Tu¯T,δTFp¯Tv¯T)F,

where, for all FT, we let κF:=𝜿T𝒏TF𝒏TF and the face-based residual operator

δTFp¯T:U¯Tp¯TpF(F)

is such that, for all v¯TU¯Tp¯T,

(3.5)δTFp¯Tv¯T:=πFpF(vF-rTpT+1v¯T+πTpTrTpT+1v¯T-vT).

The first contribution in aT is in charge of consistency, whereas the second ensures stability by a least-square penalty of the face-based residuals δTFp¯T. This subtle form for δTFp¯T ensures that the residual vanishes when its argument is the interpolate of a function in pT+1(T), and is required for high-order h-convergence (a detailed motivation is provided in [18, Remark 6]).

The global space of DOFs and its subspace with strongly enforced boundary conditions are defined, respectively, as

U¯hp¯h:=(×T𝒯hpT(T))×(×FhpF(F)),U¯h,0p¯h:={v¯hU¯hp¯hvF0 for all Fhb}.

Note that interface DOFs in U¯hp¯h are single-valued, meaning that they match from one element to the adjacent one. We use the notation

v¯h=((vT)T𝒯h,(vF)Fh)

for a generic DOF vector in U¯hp¯h and, for all T𝒯h, we denote by v¯TU¯Tp¯T its restriction to T. For further use, we also introduce the global interpolator

I¯hp¯h:H1(Ω)U¯hp¯h

such that, for all vH1(Ω),

(3.6)I¯hp¯hv:=((πTpTv)T𝒯h,(πFpFv)Fh),

and denote by I¯Tp¯T its restriction to T𝒯h.

The hp-HHO discretization of problem (1.2) consists in seeking u¯hU¯h,0p¯h such that

(3.7)ah(u¯h,v¯h)=lh(v¯h)for all v¯hU¯h,0p¯h,

where the global bilinear form ah on U¯hp¯h×U¯hp¯h and the linear form lh on U¯hp¯h are assembled element-wise setting

ah(u¯h,v¯h):=T𝒯haT(u¯T,v¯T),lh(v¯h):=T𝒯h(f,vT)T.

### Remark 3.1 (Static Condensation).

Using a standard static condensation procedure, it is possible to eliminate element-based DOFs locally and solve (3.7) by inverting a system in the skeletal unknowns only. For the sake of conciseness, we do not repeat the details here and refer instead to [12, Section 2.4]. Accounting for the strong enforcement of boundary conditions, the size of the system after static condensation is

(3.8)Ndof=Fhi(pF+d-1pF).

### Remark 3.2 (Finite Element Interpretation).

A finite element interpretation of the scheme (3.7) is possible following the extension proposed in [12, Remark 3] of the ideas originally developed in [4] in the context of nonconforming virtual element methods. For all Fhi, we denote by []F the usual jump operator (the sign is irrelevant), which we extend to boundary faces Fhb setting [φ]F:=φ. Let

𝔘h,0p¯h:={𝔳hL2(Ω)𝔳h|T𝔘Tp¯T for all T𝒯h and πFpF([𝔳T]F)=0 for all Fh},

where, for all T𝒯h, we have introduced the local space

𝔘Tp¯T:={𝔳TH1(T)(𝜿T𝔳T)pT(T) and 𝜿T𝔳T|F𝒏TFpF(F) for all FT}.

It can be proved that, for all T𝒯h,

I¯Tp¯T:𝔘Tp¯TU¯Tp¯T

is an isomorphism. Thus, the triplet (T,𝔘Tp¯T,I¯Tp¯T) defines a finite element in the sense of Ciarlet [11]. Additionally, problem (3.7) can be reformulated as the nonconforming finite element method: Find 𝔲h𝔘h,0p¯h such that

𝔞h(𝔲h,𝔳h)=lh(𝔳h)for all 𝔳h𝔘h,0p¯h,

where

𝔞h(𝔲h,𝔳h):=ah(I¯hp¯h𝔲h,I¯hp¯h𝔳h),lh(𝔳h):=lh(I¯hp¯h𝔳h),

and it can be proved that 𝔲h is the unique element of 𝔘h,0p¯h such that u¯h=I¯hp¯h𝔲h with u¯h unique solution to (3.7).

### 3.2 Main Results

We next state our main results. The proofs are postponed to Section 4. For all T𝒯h, we denote by a,T and ||s,T the seminorms defined on U¯Tp¯T by the bilinear forms aT and sT, respectively, and by a,h the seminorm defined by the bilinear form ah on U¯hp¯h. We also introduce the penalty seminorm ||s,h such that, for all v¯hU¯hp¯h,

|v¯h|s,h2:=T𝒯h|v¯T|s,T2.

Note that a,h is a norm on the subspace U¯h,0p¯h with strongly enforced boundary conditions (the arguments are analogous to those of [17, Proposition 5]). We will also need the global reconstruction operator

rhp¯h:U¯hp¯hL2(Ω)

such that, for all v¯hU¯hp¯h,

(rhp¯hv¯h)=|TrTpT+1v¯Tfor all T𝒯h.

Finally, for the sake of conciseness, throughout the rest of the paper we denote by ab the inequality aCb with real number C>0 independent of h, p¯h, 𝜿 and, for local inequalities on a mesh element T𝒯h, also of T.

Our first estimate concerns the error measured in energy-like norms.

### Theorem 3.3 (Energy Error Estimate).

Let uU and u¯hU¯h,0p¯h denote the unique solutions of problems (1.2) and (3.7), respectively, and set

(3.9)u¯^h:=I¯hp¯hu.

Assuming that u|THpT+2(T) for all TTh, it holds

(3.10)u¯h-u¯^ha,h(T𝒯hκ¯Tλ𝜿,ThT2(pT+1)(pT+1)2pTupT+2,T2)1/2.

Consequently, denoting by h the broken gradient on Th, we have that

(3.11)𝜿1/2h(u-rhp¯hu¯h)2+|u¯h|s,h2T𝒯hκ¯Tλ𝜿,ThT2(pT+1)(pT+1)2pTupT+2,T2.

### Proof.

See Section 4.3. ∎

In (3.10) and (3.11), we observe the characteristic improved h-convergence of HHO methods (cf. [18]), whereas, in terms of p-convergence, we have a more standard scaling as (pT+1)-pT (i.e., half a power more than discontinuous Galerkin methods based on polynomials of degree pT, cf., e.g., [28]). In (3.11), we observe that the left-hand side has the same convergence rate (both in h and in p) as the interpolation error

𝜿1/2h(u-rhp¯hu¯^h)2+|u¯^h|s,h2,

as can be verified combining (4.2) and (4.4) below. Note that, in this case, the p-convergence is limited by the second term, which measures the discontinuity of the potential reconstruction at interfaces. An inspection of formulas (3.10) and (3.11) also shows that the method is fully robust with respect to the heterogeneity of the diffusion coefficient, while only a moderate dependence (with a power of 12) is observed with respect to its local anisotropy ratio.

For the sake of completeness, we also provide an estimate of the L2-error between the piecewise polynomial fields uh and u^h such that

uh:=|TuTandu^h:=|Tu^T=πTpTufor all T𝒯h.

To this end, we need elliptic regularity in the following form: For all gL2(Ω), the unique element zU such that

(3.12)(𝜿z,v)=(g,v)for all vU,

satisfies the a priori estimate

(3.13)z2κ¯-1gL2(Ω),κ¯:=minT𝒯hκ¯T.

The following result is proved in Section 4.4.

Figure 1

Convergence with p-refinement (regular test case).

### (a)

u¯h-u¯^ha,h vs. Ndof

uh-u^h vs. Ndof

### (c)

u¯h-u¯^ha,h vs. p

### (d)

uh-u^h vs. p

Figure 2

Convergence with p-refinement (Le Potier’s test case).

### (a)

u¯h-u¯^ha,h vs. Ndof

uh-u^h vs. Ndof

### (c)

u¯h-u¯^ha,h vs. p

uh-u^h vs. p

### Theorem 3.4 (L2-Error Estimate).

Under the assumptions of Theorem 3.3, and further assuming elliptic regularity (3.13) and that fHpT+ΔT(T) for all TTh with ΔT=1 if pT=0 while ΔT=0 otherwise, it holds

κ¯uh-u^hκ¯1/2λ𝜿h(T𝒯hλ𝜿,Tκ¯ThT2(pT+1)(pT+1)2pTupT+2,T2)1/2+(T𝒯hhT2(pT+2)(pT+ΔT)2(pT+2)fpT+ΔT2)1/2

with λ𝛋:=maxTThλ𝛋,T and κ¯:=maxTThκ¯T.

### 3.3 Numerical Examples

We close this section with some numerical examples. The h-convergence properties of the method (3.7) have been numerically investigated in [18, Section 4]. To illustrate its p-convergence properties, we solve on the unit square domain Ω=(0,1)2 the homogeneous Dirichlet problem with exact solution u=sin(πx1)sin(πx2) and right-hand side f chosen accordingly. We consider two values for the diffusion coefficients:

𝜿1=𝑰2,𝜿2=((x2-x¯2)2+ϵ(x1-x¯1)2-(1-ϵ)(x1-x¯1)(x2-x¯2)-(1-ϵ)(x1-x¯1)(x2-x¯2)(x1-x¯1)2+ϵ(x2-x¯2)2),

where 𝑰2 denotes the identity matrix of dimension 2, 𝒙¯:=-(0.1,0.1), and ϵ=110-2. The choice 𝜿=𝜿1 (“regular” test case) yields a homogeneous isotropic problem, while the choice 𝜿=𝜿2 (“Le Potier’s” test case [27]) corresponds to a highly anisotropic problem where the principal axes of the diffusion tensor vary at each point of the domain. Figures 1 and 2 depict the energy- and L2-errors as a function of the number of skeletal DOFs Ndof (cf. (3.8)) when pF=p for all Fh and p{0,,9} for the proposed choices for 𝜿 on the meshes of Figure 3. In all the cases, the expected exponentially convergent behavior is observed. Interestingly, the best performance in terms of error vs. Ndof is obtained for the Cartesian and Voronoi meshes. A comparison of the results for the two values of the diffusion coefficients allows to appreciate the robustness of the method with respect to anisotropy.

Figure 3

Meshes considered in the p-convergence test of Section 3.3. The triangular, Cartesian, refined, and staggered meshes originate from the FVCA5 benchmark [26]; the hexagonal mesh was originally introduced in [20]; the Voronoi mesh was obtained using the PolyMesher algorithm of [33].

Triangular

Cartesian

Refined

Staggered

Hexagonal

Voronoi

## 4 Convergence Analysis

In this section we prove the results stated in Section 3.2.

### 4.1 Consistency of the Potential Reconstruction

Preliminary to the convergence analysis is the study of the approximation properties of the potential reconstruction rTpT+1 defined by (3.2) when its argument is the interpolate of a regular function. Let a mesh element T𝒯h be fixed. For any integer l1, we define the oblique elliptic projector ϖ𝜿,Tl:H1(T)l(T) such that, for all vH1(T), (ϖ𝜿,Tlv-v,1)T=0 and it holds

(4.1)(𝜿T(ϖ𝜿,Tlv-v),w)T=0for all wl(T).

### Proposition 4.1 (Characterization of rTpT+1∘I¯Tp¯T).

It holds, for all TTh,

rTpT+1I¯Tp¯T=ϖ𝜿,TpT+1.

### Proof.

For a generic vH1(T), letting v¯T=I¯Tp¯Tv in (3.2), we infer, for all wpT+1(T),

(𝜿T(rTpT+1I¯Tp¯T)v,w)T=-(πTpTv,(𝜿Tw))T+FT(πFpFv,𝜿Tw𝒏TF)F
=-(v,(𝜿Tw))T+FT(v,𝜿Tw𝒏TF)F
=(𝜿Tv,w)T,

where we have used the fact that (𝜿Tw)pT-1(T)pT(T) and (𝜿Tw)|F𝒏TFpT(F)pF(F) (cf. definition (3.1) of pT) to cancel the projectors in the second line, and an integration by parts to conclude. ∎

We next study the approximation properties of ϖ𝜿,Tl, from which those of (rTpT+1I¯Tp¯T) follow in the light of Proposition 4.1.

### Lemma 4.2 (Approximation Properties of ϖκ,Tl).

For all integers l1, all mesh elements TTh, all 0sl, and all vHs+1(T), it holds

𝜿T1/2(v-ϖ𝜿,Tlv)T+hT1/2l𝜿T1/2(v-ϖ𝜿,Tlv)T+κ¯T1/2hTv-ϖ𝜿,TlvT+κ¯T1/2hT1/2v-ϖ𝜿,TlvT
(4.2)κ¯T1/2hTmin(l,s)lsvs+1,T.

### Proof.

By definition (4.1) of ϖ𝜿,Tl, it holds

(4.3)𝜿T1/2(v-ϖ𝜿,Tlv)T2=minwl(T)𝜿T1/2(v-w)T2κ¯T(v-ΠTlv)T2.

Hence, using (2.5) with q=1, it is readily inferred that

𝜿T1/2(v-ϖ𝜿,Tlv)Tκ¯T1/2hTmin(l,s)lsvs+1,T.

To prove the second bound in (4.2), use the triangle inequality to infer

𝜿T1/2(v-ϖ𝜿,Tlv)T𝜿T1/2(v-ΠTlv)T+𝜿T1/2(ΠTlv-ϖ𝜿,Tlv)T=:𝔗1+𝔗2.

For the first term, the multiplicative trace inequality (2.3) combined with (2.5) (with q=1,2) gives

𝔗1κ¯T1/2hTmin(l,s)-1/2ls-1/2vs+1,T.

For the second term, we have

𝔗2lhT1/2𝜿T1/2(ΠTlv-ϖ𝜿,Tlv)T
lhT1/2(𝜿T1/2(ΠTlv-v)T+𝜿T1/2(v-ϖ𝜿,Tlv)T)
κ¯T1/2lhT1/2(ΠTlv-v)T
κ¯T1/2hTmin(l,s)-1/2ls-1vs+1,T,

where we have used the discrete trace inequality (2.6) in the first line, the triangle inequality in the second line, the estimate (4.3) in the third, and the approximation result (2.5) with q=1 to conclude. To obtain the third bound in (4.2), after recalling that (v-ϖ𝜿,Tlv,1)T=0, we apply the local Poincaré inequality (2.4) to infer

v-ϖ𝜿,TlvThTκ¯T1/2𝜿T1/2(v-ϖ𝜿,Tlv)Tκ¯T1/2κ¯T1/2hTmin(l,s)+1lsvs+1,T,

where the conclusion follows from the first bound in (4.2). Finally, to obtain the last bound, we use the multiplicative trace inequality (2.3) to infer

v-ϖ𝜿,TlvT2κ¯T-1/2v-ϖ𝜿,TlvT𝜿T1/2(v-ϖ𝜿,Tlv)T+hT-1v-ϖ𝜿,TlvT2,

and use the first and third bound in (4.2) to estimate the various terms. ∎

### 4.2 Consistency of the Stabilization Term

The consistency properties of the stabilization bilinear form sT defined by (3.4) are summarized in the following lemma.

### Lemma 4.3 (Consistency of the Stabilization Term).

For all TTh, all 0qpT, and all vHq+2(T), it holds

(4.4)|I¯Tp¯Tv|s,Tκ¯T1/2λ𝜿,T1/2hTmin(pT,q)+1(pT+1)qvq+2,T.

### Proof.

Let T𝒯h and vHq+2(T) and set, for the sake of brevity,

vˇT:=(rTpT+1I¯Tp¯T)v=ϖ𝜿,TpT+1v

(cf. Proposition 4.1). For all FT, recalling the definitions of the face residual δTFp¯T (cf. (3.5)) and of the local interpolator I¯Tp¯T (cf. (3.6)), together with the fact that pTpF by definition (3.1), we get

δTFp¯TI¯Tp¯Tv=πFpF(vˇT-v)-πTpT(vˇT-v).

Using the triangle inequality and the L2(F)-stability of πFpF, we infer,

(4.5)δTFp¯TI¯Tp¯TvFvˇT-vF+πTpT(vˇT-v)F=:𝔗1+𝔗2.

For the first term, the approximation properties (4.2) of ϖ𝜿,TpT+1 (with l=pT+1 and s=q+1) readily yield

(4.6)𝔗1λ𝜿,T1/2hTmin(pT,q)+3/2(pT+1)qvq+2,T.

For the second term, on the other hand, the discrete trace inequality (2.6) followed by the L2(T)-stability of πTpT and (4.2) (with l=pT+1 and s=q+1) gives

(4.7)𝔗2(pT+1)hT1/2πTpT(vˇT-v)T(pT+1)hT1/2vˇT-vTλ𝜿,T1/2hTmin(pT,q)+3/2(pT+1)qvq+2,T.

The bound (4.4) follows using (4.6)–(4.7) in the right-hand side of (4.5), squaring the resulting inequality, multiplying it by κF/hT, summing over FT, and using the bound (2.2) on card(T). ∎

### Proof of Theorem 3.3.

We start by noting the following abstract error estimate:

(4.8)u¯h-u¯^ha,hsupv¯hU¯h,0p¯h,v¯ha,h=1Eh(v¯h),

with nonconformity error

(4.9)Eh(v¯h):=lh(v¯h)-ah(u¯^h,v¯h).

To prove (4.8), it suffices to observe that

u¯h-u¯^ha,h2=ah(u¯h-u¯^h,u¯h-u¯^h)
=ah(u¯h,u¯h-u¯^h)-ah(u¯^h,u¯h-u¯^h)
=lh(u¯h-u¯^h)-ah(u¯^h,u¯h-u¯^h),

where we have used the definition of the a,h-norm in the first line, the linearity of ah in its first argument in the second line, and the discrete problem (3.7) in the third. The conclusion follows dividing both sides by u¯h-u¯^ha,h, using linearity, and passing to the supremum.

We next bound the nonconformity error Eh(v¯h) for a generic vector of DOFs v¯hU¯h,0p¯h. A preliminary step consists in finding a more appropriate rewriting for Eh(v¯h). Observing that f=-(𝜿u) a.e. in Ω, integrating by parts element-by-element, and using the continuity of the normal component of 𝜿u across interfaces together with the strongly enforced boundary conditions in U¯h,0p¯h to insert vF into the second term in parentheses, we infer

(4.10)lh(v¯h)=T𝒯h((𝜿Tu,vT)T+FT(𝜿Tu𝒏TF,vF-vT)F).

Setting, for the sake of conciseness (cf. Proposition 4.1),

(4.11)uˇT:=rTpT+1u¯^T=ϖ𝜿,TpT+1u,

and using the definition (3.2) of rTpT+1v¯T with w=uˇT, we have

(4.12)ah(u¯^h,v¯h)=T𝒯h((𝜿TuˇT,vT)T+FT(𝜿TuˇT𝒏TF,vF-vT)F+sT(u¯^T,v¯T)).

Subtracting (4.12) from (4.10), and observing that the first terms inside the summations cancel out owing to (4.1), we have

(4.13)Eh(v¯h)=T𝒯h(FT(𝜿T(uˇT-u)𝒏TF,vF-vT)F+sT(u¯^T,v¯T)).

Denote by 𝔗1(T) and 𝔗2(T) the two summands in parentheses. Using the Cauchy–Schwarz inequality followed by the approximation properties (4.2) of uˇT (with l=s=pT+1) and (4.16) below, we have for the first term

|𝔗1(T)|hT1/2𝜿T1/2(uˇT-u)T×(FTκFhTvF-vTF2)1/2
(4.14)κ¯T1/2λ𝜿,T1/2hTpT+1(pT+1)pTupT+2,Tv¯Ta,T.

For the second term, the Cauchy–Schwarz inequality followed by (4.4) (with q=pT) readily yields

(4.15)|𝔗2(T)|κ¯T1/2λ𝜿,T1/2hTpT+1(pT+1)pTupT+2,T|v¯T|s,Tκ¯T1/2λ𝜿,T1/2hTpT+1(pT+1)pTupT+2,Tv¯Ta,T.

Using (4.14)–(4.15) to estimate the right-hand side of (4.13), applying the Cauchy–Schwarz inequality, and passing to the supremum yields (3.10). To prove (3.11), it suffices to observe that, inserting u¯^h and using the triangle inequality, we obtain

𝜿1/2h(u-rhp¯hu¯h)2+|u¯h|s,h2𝜿1/2h(u-rhp¯hu¯^h)2+|u¯^h|s,h2+u¯h-u¯^ha,h2.

and (3.11) follows using the estimates (4.2), (4.4), and (3.10) to bound the terms in the right-hand side. ∎

### Remark 4.4 (Nonconformity Error).

The name nonconformity error for the quantity (4.9) is reminiscent of the Finite Element literature. As a matter of fact Eh corresponds to the contribution that appears in the estimate provided by the Second Strang Lemma when nonconforming methods are considered; see, e.g., [23, Lemma 2.25] and, in particular, the third term in the right-hand side of [23, (2.20)].

### Proposition 4.5 (Estimate of Boundary Difference Seminorm).

It holds, for all v¯TU¯Tp¯T,

(4.16)FTκFhTvF-vTF2λ𝜿,Tv¯Ta,T2.

### Proof.

Let T𝒯h, v¯TU¯Tp¯T, and set, for the sake of brevity, vˇT:=rTpT+1v¯T. We have, for all FT,

vF-vTF=πFpF(vF-vT)F
=πFpF(vF-vˇT+πTpTvˇT-vT+vˇT-πTpTvˇT)F
(4.17)δTFp¯Tv¯TF+vˇT-πTpTvˇTF,

where we have used the fact that pTpF (cf. (3.1)) to infer that vT|FpF(F) and thus insert πFpF in the first line, added and subtracted (vˇT-πTpTvˇT) in the second line, used the triangle inequality together with the definition (3.5) of the face-based residual δTFp¯T and the L2(F)-stability of πFpF in the third. To conclude, we observe that, if pT=0, then the discrete trace inequality (2.6) followed by Poincaré’s inequality yield

vˇT-πT0vˇTFhT1/2κ¯T-1/2𝜿vˇTT

while, if pT1, then

vˇT-πTpTvˇTF=vˇT-πT0vˇT-πTpT(vˇT-πT0vˇT)F
pT+1hT1/2vˇT-πT0vˇT-πTpT(vˇT-πT0vˇT)T
pT+1hT1/2hTpTvˇT-πT0vˇT1,T
hT1/2κ¯T-1/2𝜿T1/2vˇTT,

where we have inserted ±πT0vˇT in the first line, used the discrete trace inequality (2.6) in the second line, the L2(T)-optimality of πTpT together with the approximation properties (2.5) (with l=pT and q=s=0) in the third line, and we have concluded observing that (pT+1)/pT2 and using the local Poincaré inequality (2.4) to infer

vˇT-πT0vˇT1,ThT1/2vˇTT.

Plugging the above bounds for vˇT-πTpTvˇTF into (4.17), squaring the resulting inequality, multiplying it by κF/hT, summing over FT, and recalling the bound (2.2) on card(T), we deduce (4.16). ∎

### Proof of Theorem 3.4.

We let zU solve (3.12) with g=u^h-u and set z¯^h:=I¯hp¯hz and, for all T𝒯h (cf. Proposition 4.1),

(4.18)zˇT:=rTpT+1z¯^T=ϖ𝜿,TpT+1z.

For the sake of brevity, we also let e¯h:=u¯^h-u¯hU¯h,0p¯h (recall the definition (3.9) of u¯^h), so that u^T-uT=eT for all T𝒯h. We start by observing that

(4.19)eh2=-((𝜿z),eh)=T𝒯h((𝜿Tz,eT)T+FT(𝜿Tz𝒏TF,eF-eT)F),

where we have used the fact that -(𝜿z)=eh a.e. in Ω followed by element-by-element partial integration together with the continuity of the normal component of 𝜿Tz across interfaces and the strongly enforced boundary conditions in U¯h,0p¯h to insert eF into the last term.

In view of adding and subtracting ah(e¯h,z¯^h) to the right-hand side of (4.19), we next provide two useful reformulations of this quantity. First, we have

ah(e¯h,z¯^h)=ah(u¯^h,z¯^h)-ah(u¯h,z¯^h)+(f,z)-(𝜿u,z)
=T𝒯h((𝜿TuˇT,zˇT)T-(𝜿Tu,z)T+sT(u¯^T,z¯^T)+(f,z-πTpTz)T)
(4.20)=T𝒯h((𝜿T(uˇT-u),(zˇT-z))T+sT(u¯^T,z¯^T)+(f-πTpTf,z-πT1-ΔTz)T),

where we have added the quantity (f,z)-(𝜿u,z)=0 (cf. (1.2)) in the first line, we have passed to the second line using the definition (3.3) of aT (with u¯T=u¯^T and v¯T=z¯^T) together with the discrete problem (3.7) to infer

ah(u¯h,z¯^h)=(f,z^h)=T𝒯h(f,πTpTz)T,

and we have concluded using the definitions (4.1) of ϖ𝜿,TpT+1 (together with (4.11) and (4.18)) and (2.1) of πTpT and πT1-ΔT. Second, using the definition (3.2) of rTpT+1 (with v¯T=e¯T), we obtain

(4.21)ah(e¯h,z¯^h)=T𝒯h((𝜿Tz,eT)T+FT(𝜿TzˇT,eF-eT)F+sT(z¯^T,e¯T)),

where we have additionally used the fact that zˇT=ϖ𝜿,TpT+1z (cf. (4.18)) together with the definition (4.1) of ϖ𝜿,TpT+1 to replace zˇT by z in the first term in parentheses.

Thus, adding (4.20) and subtracting (4.21) from (4.19), we obtain after rearranging

(4.22)eh2=T𝒯h(𝔗1(T)+𝔗2(T)+𝔗3(T)),

with

𝔗1(T):=FT(𝜿T(z-zˇT)𝒏TF,eF-eT)F+sT(z¯^T,e¯T),
𝔗2(T):=(𝜿T(uˇT-u),(zˇT-z))T+sT(u¯^T,z¯^T),
𝔗3(T):=(f-πTpTf,z-πT1-ΔTz)T.

Using the Cauchy–Schwarz inequality, the approximation properties (4.2) of zˇT (with l=pT+1 and s=1) together with the consistency properties (4.4) of sT (with q=0) for the first factor, and the bound (4.16) for the second factor, we get

|𝔗1(T)|(hT𝜿T1/2(z-zˇT)T2+|z¯^T|s,T2)1/2×(FTκFhTeF-eTF2+|e¯T|s,T2)1/2
(4.23)κ¯T1/2λ𝜿,ThTe¯Ta,Tz2,T.

For the second term, the Cauchy–Schwarz inequality followed by the approximation properties (4.2) of uˇT (with q=pT) and zˇT (with q=1), and the consistency properties (4.4) of sT (with q=pT and q=0 for the first and second factor, respectively) yield

|𝔗2(T)|(𝜿1/2(uˇT-u)T2+|u¯^T|s,T2)1/2×(𝜿1/2(zˇT-z)T2+|z¯^T|s,T2)1/2
(4.24)κ¯Tλ𝜿,ThTpT+2(pT+1)pTupT+2,Tz2,T.

Finally, for the third term we have, when pT=0,

(4.25)|𝔗3(T)|f-πT0fTz-πT0zThT2f1,Tz1,ThT2f1,Tz2,T,

while, when pT1,

|𝔗3(T)|f-πTpTfTz-πT1zT
f-ΠTpTfTz-ΠT1zT
(4.26)hTpT+2pTpT+2fpTz2,ThTpT+2pTpT+2fpT,Tz2,T,

where we have used the optimality of πTpT in the L2(T)-norm to pass to the second line and the approximation properties (2.5) of ΠTpT to conclude. Using (4.23)–(4.26) to bound the right-hand side of (4.22), and recalling the energy error estimate (3.10) and elliptic regularity (3.13) concludes the proof. ∎

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: project HHOMM (ANR-15-CE40-0005)

Funding statement: We acknowledge the support of Agence Nationale de la Recherche through project HHOMM (ANR-15-CE40-0005).

## A Proof of Lemma 2.3

Let K^d be a L-Lipschitz set (that is, such that its boundary can be locally parametrized by means of L-Lipschitz functions) with diam(K^)=1, and fix r0>1 and a d-cube R(r0) containing K^. In [6, Lemma 3.1], the following is shown: Suppose that K^ is a cube, then given a function vHs+1(K^), there exists a polynomial ΠK^lvl(K^) that satisfies

(A.1)v-ΠK^lvq,K^1ls+1-qvs+1,K^

for qs+1. Actually, by analyzing the proof of that result, we can reinforce that statement not only for cubes, but for sets K^ which are regular enough, in the sense that there has to exist an extension operator E:Hs+1(K^)Hs+1(R(2r0)) such that

(A.2)E(v)s+1,R(2r0)Cvs+1,R(K^),E(v)=0 on R(2r0)R(32r0).

So the problem is to show the existence of such an extension (in any dimension d1). The minimal assumptions such that this is doable are studied in [32, Theorems 5 and 5, p. 181]. Namely, by means of a careful inspection of [32, Theorems 5 and 5] and their proofs, and in particular formulas (25), (30) and the end of the proof of Theorem 5 at p. 192, we get that the constant C in (A.2) depends on the Lipschitz constant L and on the (minimal) number of L-Lipschitz coverings of K^, that is, the number of open sets which cover K^ and in each of whom K^ can be parametrized by means of an L-Lipschitz function.[1] Thus, to get the hp-estimate (2.5), we have to show two facts:

1. first, that (2.5) follows from (A.1) whenever we replace K^ with an element T of the mesh;

2. second, that (A.1) actually holds with a uniform bounding constant for the elements of the mesh.

(i) Proof of (2.5) for regular elements. Assume for the moment that any T𝒯h satisfies the regularity assumptions needed to have the extension (A.2), and thus (A.1), with T^:=T/hT in place of K^. We additionally assume, without loss of generality, that the barycenter of T (and thus of T^) is . Then, by homogeneity, we get that, for every fHr(T), letting f^(𝒙):=f(𝒙/λ),

(A.3)fr,Tλd/2-rf^r,Tλ.

Thus, setting r=s+1, λ=hT, and f=v-𝔮 with 𝔮 a generic polynomial of degree l, we get by (A.1) (applied to v-𝔮 and ΠTl(v-𝔮) in place of v and ΠTlv, respectively),

v-ΠTlvq,T=(v-𝔮)-ΠTl(v-𝔮)q,ThTd/2-qls+1-qv^-𝔮^s+1,T^.

Now, [22, Theorem 3.2] states that

|f-ΠT^lf|i,T^|f|l,T^

for any il+1. Thus by the definition of the Hq-norm we conclude that

v-ΠTlvq,ThTd/2-qls+1-q(i=min(l,s)+1s+1|v^|i,T^2)1/2hTmin(l,s)-q+1ls+1-qvs+1,T.

Notice that we exploited again (A.3) to scale back on T, and the fact that hT<1.

(ii) Proof of regularity under Assumption 2.1. To conclude the proof, we are left to show that Assumption 2.1 entails a uniform bound only in terms of ϱ for the Lipschitz constant of every element T𝒯h. To this aim, consider 𝒙T. Then, 𝒙S for some (convex) element of the submesh S𝔗h contained in T. Since the sets S form a submesh of 𝒯h, a uniform bound on the Lipschitz regularity of S would imply a bound on the Lipschitz regularity of T. Thus, we focus on the regularity of S. Since S is convex, we can cover S by means of 2(d+1) open sets Ui, such that SUi admits a local convex (and thus Lipschitz) parametrization ϕi, i.e., there exists an orthogonal coordinate system such that SUi is the graph of a Lipschitz function ϕi:Iid-1. This bound on the number of open sets Ui is crucial to get [32, Theorem 5] to work (clearly, thanks to (2.2), the bound on the number of Lipschitz coverings of T is bounded by a constant 2dN=c(d,ϱ)). We claim that each ϕi is 1/ϱ-Lipschitz.

Suppose that 𝒙Ui=:U and set ϕ:=ϕi. Up to a rotation and a rescaling, we can suppose that 𝒙= and ϕ(𝒙)=ϕ()=0. Let now rs be the inradius of S and hS be its diameter. By Assumption 2.1, we know that hsrS1ϱ. Let Brs be a ball contained in S of radius rS. Up to a further rotation of center 𝒙= of the coordinate system, we can suppose that BrS is centered on the xd-axis. In place of ϕ:I, it is useful to consider its Lipschitz extension ϕ~:d-1 defined by, denoting by || the usual Euclidian norm,

ϕ~(𝒙):=inf{ϕ(𝒚)+Lip(ϕ)|𝒚-𝒙|𝒚d-1},

We know that ϕ~ is Lipschitz on d-1 and that Lip(ϕ~)=Lip(ϕ) (see for instance [2, Proposition 2.12]). Moreover, it is clear that ϕ~ is convex on d-1. The fact that BrSS and it is centered on the xd-axis (without loss of generality, we can suppose that its center is 𝝃=(,ξd) with ξd>0) translates into the fact that BrS is contained in the epigraph of ϕ~ and its center has distance from d at most hS. Let now 𝒑ϕ~(0), where ϕ~ is the subdifferential of ϕ~. Then, for every 𝒚d-1 we have

ϕ~(𝒚)𝒑𝒚.

By choosing 𝒚=λ𝒑, with λ0, we get the inequality

(A.4)ϕ~(λ𝒑)λ|𝒑||𝒑|.

Since the epigraph of ϕ~ contains BrS, which is centered at a height less than hS on the xd-axis, and by the convexity of ϕ~, we have that the truncated cone

C={(𝒙,xn)d-1×hSxnhSrS|𝒙|}

is contained in the epigraph of ϕ~ (see Figure 4). Then from (A.4) we get

|𝒑|2ϕ~(𝒑)hSrS|𝒑|

and so, by Assumption 2.1,

|𝒑|hSrS1ϱ.

Let now 𝒚d-1. Then

|ϕ~(𝒚)-ϕ~(𝒙)||𝒑(𝒚-𝒙)||𝒑||𝒚-𝒙|ϱ-1|𝒚-𝒙|.

Since 𝒙 is arbitrary, this shows that Lip(ϕ)=Lip(ϕ~)=ϱ-1, and so that S is ϱ-1-Lipschitz.

### Figure 4

Illustration for point (ii) in the proof of Lemma 2.3.

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text:
# How do transmission lines affect the impedance of some arbitrary load?

Let's say I have something like a non-resonant dipole, so that impedance changes depending on frequency and could take any number of values.

Whatever that impedance is, how does adding transmission lines or baluns change the impedance seen by the transceiver?

You are probably familiar with SWR, where 1:1 represents a perfect match. SWR quantifies the magnitude of mismatch between two impedances, such as the antenna and the transmission line. As the two impedances become more unequal, the SWR increases.

The SWR does not change with the length of the transmission line, as long as that transmission line is lossless. Usually transmission losses are low enough to be negligible for impedance calculations.

SWR tells you the magnitude of mismatch, but nothing about the relative impedances. For example, 25Ω and 200Ω are both a 2:1 SWR to 50Ω coax. There are also any number of complex impedances that are also a 2:1 SWR.

There is a more general concept to represent mismatches: reflection coefficient ($\Gamma$). It's a complex number that quantifies not only the magnitude of the mismatch, but also the phase.

I find it easiest to visualize on a Smith Chart. Let's say we have a 300Ω ladder line, and a load (an antenna, say) with an (arbitrarily selected) impedance of (90+150j)Ω. In the Smith Chart, the center represents our target impedance (in this case, 300Ω) and all the impedances are normalized to that value. So we divide (90+150j)Ω by 300 to get (0.3+0.5j)Ω, and plot that on the Smith Chart:

By measuring the distance from the center, and the angle, we get the reflection coefficient in polar coordinates.

That's the impedance of the antenna at the feedpoint, normalized to 300Ω. Now let's add a 300Ω transmission line of increasing length, and consider the impedance at the end of it that would be seen, for example by the transmitter.

As the transmission line gets longer, the equivalent impedance moves in a clockwise circle around the center of the chart. This changes the phase of the reflection coefficient, but not the magnitude. Since the magnitude is constant, so is the SWR.

The distance moved around that circle depends on how long the transmission line is. When its a half-wavelength, it will have gone a full circle. A full circle is just a half-wavelength because the reflection must travel over that half-wavelengh twice: there and back.

We can stop at any time and read our new position on the Smith chart. Then, multiply the values read from the Smith chart by the normalized impedance (300Ω in our example). In this example, we end up at (0.28-0.43j)Ω. Multiplied by 300 makes (84-129j)Ω.

What if now we add a 4:1 balun? Depending on which way we orient the balun, we can step the impedance up or down by 4. Let's step down: simply divide the impedance by 4 to get (21-32.25)Ω.

Now, let's add some 50Ω coax. Divide by 50 to get a normalized impedance of (0.42-0.645)Ω. Plot that:

As before, as the transmission line gets longer, the dot moves about the center in a clockwise circle. Multiply the final location by 50 to get the equivalent impedance seen by the transmitter.

As luck would have it, for this 50Ω coax we've ended up a little closer to the center, so the 50Ω coax will have a lower SWR than the 300Ω ladder line. I've just been picking arbitrary impedances and lengths, but with the careful selection of transmission line impedances and lengths this can actually be used to match loads.

• This is wonderful. Thanks for taking the time to put this together. – bcattle Feb 4 '17 at 4:31

The answer given by Phil Frost - W8II is comprehensive. Using Smith Chart makes it graphical and very easy to understand and very easy to use.

I am giving below the same answer, but in a mathematical form, in case some one is interested.

An example of use of phenomenon "adding transmission line changes the impedance seen by the transceiver" to match impedance of antenna to impedance of receiver

Receiver and coax impedance = 75 ohms
Antenna impedance = 126 - j38 ohms
Antenna is designed for 1090 MHz, λ = 275 mm
Velocity Factor of Coax = 0.83, 1/2 λ = 1/2 x 275 x 0.83 = 114 mm

Method:
Using Smith Chart, determine the length of Coax where resistance becomes 75 ohms i.e. transmission line circle intersects Smith Chart's circle R=1.0 (normalized to 75 ohms).

At this point (in this example) the reactance seen is inductive (+ve). Calculate the value of a capacitor whose capacitive reactance (-ve) is equal to inductive reactance. Placing this capacitor in series of coax core at feed point will neutralize inductive reactance and impedance seen =75+j0 ohms.

SMITH CHART


text:
# How do you differentiate y=sqrtx/(x^3+1) using the quotient rule?

##### 1 Answer
Oct 29, 2017

$\frac{1 - 5 {x}^{3}}{2 \sqrt{x} {\left({x}^{3} + 1\right)}^{2}}$

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(f \frac{x}{g} \left(x\right)\right) = \frac{g \left(x\right) \cdot \frac{d}{\mathrm{dx}} f \left(x\right) - f \left(x\right) \cdot \frac{d}{\mathrm{dx}} g \left(x\right)}{g {\left(x\right)}^{2}}$ ----------->(A)

here

f(x)=$\sqrt{x}$ and
g(x)= ${x}^{3} + 1$
$\frac{d}{\mathrm{dx}} \left(\sqrt{x}\right) = \left(\frac{1}{2 \cdot \sqrt{x}}\right)$
$\frac{d}{\mathrm{dx}} \left({x}^{3} + 1\right) = 3 \cdot {x}^{2}$

Substitute the above values in equation (A) and calculate the required answer.