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This month's product spotlight features Chilewich materials. The name is synonymous with a modern aesthetic that fuses style with function. Textile designer, Sandy Chilewich applies her interest in color and experimentation with fibers to create signature textiles. Chilewich materials are now available in window treatments for the home as well as commercial interiors through The Shade Store.
Versatile and durable, Chilewich materials have been purposed into everything from flooring and table runners to bags and dresses. You read correctly – dresses! Last Fall, Chilewich made their debut on the fashion runway for Chanel.
Our Roller Shades and Vertical Systems by Chilewich are all on sale until the end of April. You can view our best photography of the Chilewich products in our gallery.


My dog makes wheezing sounds…kennel cough?
Bailey (cairn terrier poodle) makes these weird wheezing sounds maybe twice a day. The vet gave him some antibiotics for tracheobronchitis but this doesn't sound like a kennel cough to me…. Poor thing. He also said the dog "might" have a heart murmur.


Thumbs Up for Powerpoint Backgrounds for Presentation Templates free on category Flowers! To set the image as powerpoint templates, right click the image and choose to download for microsoft powerpoint!
Licencing! Thumbs Up for Powerpoint PPT Templates - The pictures are free for personal and even for commercial use. You can modify, copy and distribute the photos on Thumbs Up for Powerpoint in PPTBackgroundsTemplates.com. All without asking for permission or setting a link to the source. So, attribution is not required.


text:
In electrochemistry, Faraday efficiency (also called faradaic efficiency, faradaic yield, coulombic efficiency or current efficiency) describes the efficiency with which charge (electrons) is transferred in a system facilitating an electrochemical reaction.  The word "Faraday" in this term has two interrelated aspects: first, the historic unit for charge is the faraday (F), but has since been replaced by the coulomb (C); and secondly, the related Faraday's constant () correlates charge with moles of matter and electrons (amount of substance).  This phenomenon was originally understood through Michael Faraday's work and expressed in his laws of electrolysis.

Sources of faradaic loss
Faradaic losses are experienced by both electrolytic and galvanic cells when electrons or ions participate in unwanted side reactions.  These losses appear as heat and/or chemical byproducts.

An example can be found in the oxidation of water to oxygen at the positive electrode in electrolysis. Some electrons are diverted to the production of hydrogen peroxide. The fraction of electrons so diverted represent a faradaic loss and vary in different apparatuses.

Even when the proper electrolysis products are produced, losses can still occur if the products are permitted to recombine.  During water electrolysis, the desired products (H2 and O2), could recombine to form water.  This could realistically happen in the presence of catalytic materials such as platinum or palladium commonly used as electrodes.  Failure to account for this Faraday-efficiency effect has been identified as the cause of the misidentification of positive results in cold fusion experiments.

Proton exchange membrane fuel cells provide another example of faradaic losses when some of the electrons separated from hydrogen at the anode leak through the membrane and reach the cathode directly instead of passing through the load and performing useful work. Ideally the electrolyte membrane would be a perfect insulator and prevent this from happening.

An especially familiar example of faradaic loss is the self-discharge that limits battery shelf-life.

Methods of measuring faradaic loss
Faradaic efficiency of a cell design is usually measured through bulk electrolysis where a known quantity of reagent is stoichiometrically converted to product, as measured by the current passed. This result is then compared to the observed quantity of product measured through another analytical method.

Faradaic loss vs. voltage and energy efficiency
Faradaic loss is only one form of energy loss in an electrochemical system. Another is overpotential, the difference between the theoretical and actual electrode voltages needed to drive the reaction at the desired rate. Even a rechargeable battery with 100% faradaic efficiency requires charging at a higher voltage than it produces during discharge, so its overall energy efficiency is the product of voltage efficiency and faradaic efficiency. Voltage efficiencies below 100% reflect the thermodynamic irreversibility of every real-world chemical reaction.

References

Electrochemistry

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redpajama_set_name: RedPajamaWikipedia


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APPI Ranch is renewed and opens as "APPI-NO-MORI" (APPI Forest) with new flower garden on Jul 2017. Come & enjoy BBQ, interactions with animals, fishing and more! Please take tea time at lunchroom after having enjoyable time.
All small animals will welcome your visit!
* There is also FREE to use outdoor Dog Runs.
Enjoy fishing the fish! You can cook the fish after fishing here.
*3 sets of pole & feed, and 5 fishes (with charcoal burner fee) are included.
Very popular Hercules Beetle and Giant Stag Beetle for kids. You can also watch and touch the beetles!
OPEN NEW LUXURY CAMPING! Please enojy it with BBQ here!
This is a lunch/rest place after enjoying APPI activities. Here are the various menus for foods and drinks.
So popular for kids! Dad & mom can also ride with kids together.
Even kids can drive here. It's fun and safe in the space.
Fun to keep jumping easily as many as possible here!
* The price shown per person 10mins.


Is there anything worse than coming home to a freezing cold house in the middle of winter because your furnace went out? At Neerings Plumbing & Heating, we don't think so. That's why we are proud to offer only the best Salt Lake City furnace repair and maintenance services. We know just how brutal winters in Utah can be, and we are here to ensure that your family stays warm and cozy no matter how low the temperatures dip.
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Unfortunately, furnace problems have a tendency to sneak up on you at the worst possible time. If your furnace stops working in the middle of the night in the dead of winter, don't worry! At Neerings Plumbing & Heating, we know how important it is to get your home warm again, so we offer emergency furnace repair services in Salt Lake City. When you call us at 801-467-3951 to ask for our help in a heating emergency, we will respond to you as quickly as possible so your family isn't left in the cold for an extended period of time.
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SUITCASE has teamed up with contemporary-cool luxury travel brand Horizn Studios to bring you jet-setting hacks from those who know best, helping you to nail the urban-nomad lifestyle wherever your next adventure is taking you. Scroll down to enter our competition for the chance to win £600 worth of Horizn luggage.
Here, we uncover the hidden costs of travelling abroad and how to avoid them – without having to create a ghastly spreadsheet of expenses.
1. Withdraw money before you go to avoid paying hefty fees (credit cards can charge up to 3%). Check to see if your bank is part of a network of international partner ATMs to reduce or avoid these surcharges, or choose a credit company with no additional costs at all.
2. Always exchange currency before you get to the airport to avoid nasty commission fees. Do your research to ensure you're getting the best exchange rate on the high street.
3. If you're road tripping, look out for sneaky car rental add-ons which you may not need. Check in with your credit company to see if they provide coverage – if so, paying on your card will mean you avoid additional fees from the rental company.
4. Driving to the airport? Book your parking space in advance, which can be up to three times cheaper than doing it in advance. Car sharing iniatives (which can be found online) are a good way to protect the environment and your bank balance.
5. Tourist or city tax – usually added onto your hotel bill and paid to the local authority – is unavoidable in most large cities (with occasional exceptions for business travellers) and can amount to a whopping 15% in places like New York – don't forget to factor this into your budget, as well as departure fees which are not always included in the price of your plane ticket.
6. Most of us want to stay connected while travelling (how else are you going to fill your social media feeds with hot-dogs-or-legs snaps?) but roaming charges are downright hideous. Contact your mobile service provider to find out about switching to a temporary international package while you're away or consider using a global SIM for your trip. Alternatively, download any apps you may need while you're away before you travel then ditch the internet and use your trip as an excuse to unplug and reset.
7. Ensure that wherever you're staying has free wifi – it will invariably end up cheaper than paying a daily charge. If it's not included, consider investing in a pocket wifi add-on so you can get online whenever, wherever.
8. Travelling by boat? You'll be cruising for a bruising if you forget about added port fees and ship docking charges – make sure you know exactly what is included in your ticket.
9. Look into transport options from the airport before you arrive to avoid the last-minute lazy taxi ride to your accommodation which is all too easy after a long-haul flight.
10. Check out the local tipping rates at your destination – it's always good to be on the generous side, but you're (probably) not made of money, and in some places (such as Japan) it's even considered rude to tip!


text:
\section{Introduction} \label{intro}
In this note we study an adaptive finite element method for the reconstruction of a dielectric permittivity function $\varepsilon=\varepsilon(\mathbf{x})$, $\mathbf{x}\in\Omega$, where $\Omega\subset\mathbb{R}^3$ is a bounded domain with (piecewise) smooth boundary $\Gamma$. This is a coefficient inverse problem (CIP) for Maxwell's equations, where the dielectric permittivity function $\varepsilon$, acting as the coefficient in the equations, characterizes an inhomogeneous, isotropic, non-magnetic, non-conductive medium in $\Omega$. Possible applications include detection of explosives in airport security and detection of land mines.

The method studied is based on minimization of a Tikhonov functional, where the functions involved are approximated by piecewise polynomials. It is intended as a second stage in a two-stage numerical procedure for the reconstruction of a dielectric permittivity. On the first stage, described in \cite{bk12, bk12b}, a good initial approximation $\varepsilon_0$ of the dielectric permittivity function is obtained by a globally convergent method. This initial approximation is then refined on the second stage.

The version of the second stage considered here was introduced in \cite{m14}. Another version was studied theoretically and numerically in \cite{b11, bkk10, bk10a, btkm14, btkm14b}. There were two main reasons for introducing the new version of the second stage in \cite{m14}. The first reason was to handle a discrepancy between theory and implementation which was present in the previous version. This discrepancy was primarily due to the fact that the dielectric permittivity was approximated by a piecewise constant function, while the theory required higher regularity. In spite of that discrepancy, reasonable reconstructions were obtained, but it remained to be seen whether the new version of \cite{m14} could produce even more accurate reconstructions.

The second reason to introduce the version of \cite{m14} was to incorporate the divergence free condition for the electric displacement directly into the differential equation, without having to introduce an additional stabilizing penalty term as was done in \cite{b11, bkk10, bk10a, btkm14, btkm14b}.

In \cite{b11}, an a posteriori error estimate for a Lagrangian functional was derived. A similar estimate was given in \cite{m14}, but there the amount of detail provided in the proof was, for the sake of brevity, kept to a minimum. Here we give the fully detailed proof of that estimate. Moreover, we extend the error analysis also to include a posteriori error estimation for the Tikhonov functional, as well as for the permittivity function itself. The arguments which we use here could easily be adapted to obtain such estimates also for the original version of the second stage considered in \cite{b11, bkk10, bk10a, btkm14, btkm14b}.

The remaining part of this note is structured as follows: In the next section we present the mathematical formulations of the direct and inverse problems and present the basic results prior to discretization of the problems. In Section~\ref{error_analysis} we state the finite element formulations and perform the error analysis. Some concluding remarks are given in Section~\ref{conclusion}.

\section{The direct and inverse problems} \label{dir_inv}
Before proceeding with the mathematical statement of the problem, we introduce some notation. For the bounded domain $\Omega\subset \mathbb{R}^3$ with boundary $\Gamma$, we write $ {\Omega_T} \coloneqq \Omega\times(0,\,T)$ and $ {\Gamma_T} \coloneqq\Gamma\times(0,\,T)$, where $T>0$ is a (sufficiently large) fixed time. If $X\subset\mathbb{R}^n$, $n\in\mathbb{N}$, is a domain, we define the norm $\norm{\cdot}_{X,\,m}\coloneqq\norm{\cdot}_{H^m(X)}$ and corresponding inner product $\sca{\cdot}{\cdot}_{X,\,m}\coloneqq\sca{\cdot}{\cdot}_{H^m(X)}$, where $H^m(X)$ is the $L_2$-based Sobolev space of order $m$ over $X$, with respect to the usual Lebesgue measure. To simplify notation, we will drop the index $m$ whenever it is zero.

Let $V^\varepsilon\coloneqq H^3(\Omega)$. We define the set of admissible dielectric permittivity functions
\begin{equation} \label{admissible}
U^\varepsilon\coloneqq\{v\in V^\varepsilon: 1 \leq v(\mathbf{x}) \leq \varepsilon_{\mathrm{max}}~\forall \mathbf{x}\in\Omega,~v\rvert_\Gamma\equiv1,~\nabla v\rvert_\Gamma\equiv 0\}
\end{equation}
for some known but not necessarily small upper bound $\varepsilon_{\mathrm{\max}}$. The set $U^\varepsilon$ is defined to describe a heterogeneous medium in $\Omega$, immersed in a constant background with permittivity 1 in $\mathbb{R}^3\setminus\Omega$.

Under the assumption that $\varepsilon\in U^\varepsilon$ we consider Maxwell's equations for an isotropic, non-magnetic, non-conductive medium in $\Omega$:
\begin{align}
\pd{(\mu\H)}{t} + \nabla\times\mathbf{E} &= 0 && \text{in }  {\Omega_T} , \label{mw1}\\
\pd{(\varepsilon\mathbf{E})}{t} - \nabla\times\H &= 0 && \text{in }  {\Omega_T} , \label{mw2}\\
 \div(\mu\H) = \div(\varepsilon\mathbf{E}) &= 0 && \text{in }  {\Omega_T} , \label{mw3}
\end{align}
where $\H=\H(\mathbf{x},\,t)$ and $\mathbf{E}=\mathbf{E}(\mathbf{x},\,t)$, $(\mathbf{x},\,t)\in {\Omega_T} $, denote the magnetic and electric fields, respectively, and $\mu>0$ is the constant magnetic permeability. By scaling, we may assume that $\mu=1$.

To obtain an equation involving only $\varepsilon$ and $\mathbf{E}$, we combine the curl of \eqref{mw1} and derivative of \eqref{mw2} with respect to $t$ to obtain the second order equation
\begin{align*}
\varepsilon\pd{^2\mathbf{E}}{t^2} + \nabla\times(\nabla\times\mathbf{E}) &= 0 && \text{in }  {\Omega_T} .
\end{align*}
To incorporate \eqref{mw3} we proceed as in \cite{m14} to expand $\nabla\times(\nabla\times\mathbf{E})=-\Delta\mathbf{E} + \nabla(\div\mathbf{E})$ and use
\begin{align*}
\div\mathbf{E} = \div\left(\frac{\varepsilon\mathbf{E}}{\varepsilon}\right) = \frac{\div(\varepsilon\mathbf{E})}{\varepsilon} - \frac{\nabla\varepsilon\cdot\mathbf{E}}{\varepsilon},
\end{align*}
where the term $\div(\varepsilon\mathbf{E})/\varepsilon$ vanishes in view of \eqref{mw3}.

Thus, after completing with boundary and initial conditions, we obtain the system
\begin{equation} \label{maxwells}
\begin{aligned}
&\varepsilon\pd{^2\mathbf{E}}{t^2}-\Delta\mathbf{E} - \nabla\left(\frac{\nabla\varepsilon\cdot\mathbf{E}}{\varepsilon}\right) = 0 &&\text{in }  {\Omega_T} ,\\
&\pd{\mathbf{E}}{\boldsymbol{\nu}} = \P &&\text{on }  {\Gamma_T} , \\
&\mathbf{E}(\cdot,\,0) = \pd{\mathbf{E}}{t}(\cdot, \,0) = 0 &&\text{in } \Omega,
\end{aligned}
\end{equation}
where $\pd{}{\boldsymbol{\nu}}=\boldsymbol{\nu}\cdot\nabla$, $\boldsymbol{\nu}$ denotes the outward unit normal on $\Gamma$, and $\P\in[L_2( {\Gamma_T} )]^3$ is given Neumann data (see Section~4 of \cite{btkm14} for details). For well-posedness of problems of this class, we refer to \cite{l85}.

The mathematical statement of the coefficient inverse problem is:

\begin{problem} \label{cip}
Given time-resolved boundary observations $\mathbf{G}\in[L_2( {\Gamma_T} )]^3$ of the electric field, determine $\varepsilon\in U^\varepsilon$ such that $\mathbf{E} = \mathbf{G}$ on $ {\Gamma_T} $.
\end{problem}

The observations $\mathbf{G}$ represents either experimental or (partially) simulated data, see \cite{btkm14}.

Uniqueness of the solution of coefficient inverse problems of this type is typically obtained via the method of Carleman estimates \cite{bk81}. Examples where this method is applied to inverse problems for Maxwell's equations can be found in, for example, \cite{k86}, \cite{bcs12} for simultaneous reconstruction of two coefficients, and \cite{li05, ly07} for bi-isotropic and anisotropic media.
However, this technique requires non-vanishing initial conditions for the underlying partial differential equation, which is not the case here. Thus, currently, uniqueness of the solution for the problem we study is not known. For the purpose of this work, we will assume that uniqueness holds. This assumption is justified by the numerical results presented in \cite{btkm14, btkm14b}.

We introduce the space $V^\mathrm{dir}\coloneqq\{\mathbf{v}\in[H^1( {\Omega_T} )]^3:\mathbf{v}(\cdot,\,0)=0\}$ for solutions to the direct problem, and $V^\mathrm{adj}\coloneqq\{\mathbf{v}\in[H^1( {\Omega_T} )]^3:\mathbf{v}(\cdot,\,T)=0\}$ for adjoint solutions. Both spaces are equipped with the usual norm and inner product on $[H^1( {\Omega_T} )]^3$.
Then, by multiplying the first equation in \eqref{maxwells} by a test function $\boldsymbol{\phi}\in V^\mathrm{adj}$ and integration over $ {\Omega_T} $, we obtain, after integration by parts,
\begin{equation} \label{weakmaxwell}
\begin{aligned}
0 &=
 -\sca{\varepsilon\pd{\mathbf{E}}{t}}{\pd{\boldsymbol{\phi}}{t}}_ {\Omega_T}
 + \sca{\varepsilon\pd{\mathbf{E}}{t}(\cdot,\,T)}{\boldsymbol{\phi}(\cdot,\,T)}_\Omega
 - \sca{\varepsilon\pd{\mathbf{E}}{t}(\cdot,\,0)}{\boldsymbol{\phi}(\cdot,\,0)}_\Omega \\&\quad
 + \sca{\nabla\mathbf{E}}{\nabla\boldsymbol{\phi}}_ {\Omega_T}
 - \sca{\pd{\mathbf{E}}{\boldsymbol{\nu}}}{\boldsymbol{\phi}}_ {\Gamma_T}
 + \sca{\frac{\nabla\varepsilon\cdot\mathbf{E}}{\varepsilon}}{\div\boldsymbol{\phi}}_ {\Omega_T}
 - \sca{\frac{\nabla\varepsilon\cdot\mathbf{E}}{\varepsilon}}{\boldsymbol{\nu}\cdot\boldsymbol{\phi}}_ {\Gamma_T} \\
&=
 -\sca{\varepsilon\pd{\mathbf{E}}{t}}{\pd{\boldsymbol{\phi}}{t}}_ {\Omega_T}
 + \sca{\nabla\mathbf{E}}{\nabla\boldsymbol{\phi}}_ {\Omega_T}
 + \sca{\frac{\nabla\varepsilon\cdot\mathbf{E}}{\varepsilon}}{\div\boldsymbol{\phi}}_ {\Omega_T}
 - \sca{\P}{\boldsymbol{\phi}}_ {\Gamma_T}  \\
& \eqqcolon \mathscr{D}(\varepsilon,\,\mathbf{E},\,\boldsymbol{\phi}),
\end{aligned}
\end{equation}
where the second equality holds because $\boldsymbol{\phi}(\cdot,\,T)=0$, $\pd{\mathbf{E}}{t}(\cdot,\,0)=0$, $\pd{\mathbf{E}}{\boldsymbol{\nu}}=\P$ on $ {\Gamma_T} $, and $\nabla\varepsilon=0$ on $ {\Gamma_T} $. This leads to the following weak description of the electric field:
\begin{problem} \label{weakdirect}
Given $\varepsilon\in U^\varepsilon$, determine $\mathbf{E}\in V^\mathrm{dir}$ such that $\mathscr{D}(\varepsilon,\,\mathbf{E},\,\boldsymbol{\phi})=0$ for every $\boldsymbol{\phi}\in V^\mathrm{adj}$.
\end{problem}

Let $\mathbf{E}_\varepsilon\in V^\mathrm{dir}$ denote the solution to Problem~\ref{weakdirect} for a given $\varepsilon\in U^\varepsilon$. We can then define the Tikhonov functional $F\colon U^\varepsilon\to \mathbb{R}_+$,
\begin{equation} \label{tikhonovfunctional}
F(\varepsilon)=F(\varepsilon,\,\mathbf{E}_\varepsilon)\coloneqq \frac{1}{2}\norm{(\mathbf{E}_\varepsilon - \mathbf{G})z_\delta}_ {\Gamma_T} ^2 + \frac{\alpha}{2}\norm{\varepsilon - \varepsilon_0}_\Omega^2,
\end{equation}

\begin{center}
  \begin{figure}
    \includegraphics[width=0.9\textwidth]{./zdelta.eps}
    \caption{Schematic illustration of the cut-off function $z_\delta$ appearing in the Tikhonov functional \eqref{tikhonovfunctional}.}\label{zdelta}
  \end{figure}
\end{center}
where $\alpha>0$ is a regularization parameter and $z_\delta=z_\delta(t)\in C^\infty([0,\,T])$ is a cut-off function for the data, dropping from a constant level of 1 to a constant level of 0 within the small interval $(T-\delta,\,T-\delta/2)$, $\delta \ll T$, as schematically shown in Figure \ref{zdelta}. The function $z_\delta$ is introduced to ensure data compatibility in the adjoint problem arising in the minimization of \eqref{tikhonovfunctional}.

How to choose the regularization parameter $\alpha$ with respect to the level of noise in the data is a widely studied topic. Several methods exist, examples are the (generalized) discrepancy principle \cite{tgsy95} and iterative methods \cite{bks11}. For the results presented here, we regard $\alpha$ as a fixed parameter.

As remarked before, the initial approximation $\varepsilon_0$ is obtained using the globally convergent method, as described in, for instance \cite{bk12b}. This means in particular that if $\varepsilon_0$ is sufficiently close to an ideal solution $\varepsilon^*$, corresponding to noiseless data $\mathbf{G}^*$, and if the regularization parameter $\alpha$ is chosen appropriately with respect to the level of noise in the data $\mathbf{G}$, then by Theorem~3.1 of \cite{bkk10}, the Tikhonov functional $F$ is strongly convex in a neighborhood $\mathcal{N}\subset V^\varepsilon$ of $\varepsilon_0$. If so, then in particular there exists a constant $c>0$ such that for every $\varepsilon_1$, $\varepsilon_2\in\mathcal{N}\cap U^\varepsilon$,
\begin{equation} \label{convexity}
c\norm{\varepsilon_1 - \varepsilon_2}^2_{V^\varepsilon} \leq F'(\varepsilon_1;\,\varepsilon_1 - \varepsilon_2) - F'(\varepsilon_2;\,\varepsilon_1 - \varepsilon_2),
\end{equation}
where $F'(\varepsilon;\,{\bar{\eps}})$ denotes the Fr\'echet derivative of $F$ at $\varepsilon$, acting on ${\bar{\eps}}$.

Throughout the remaining part of this text we will assume that the hypothesis of Theorem~3.1 of \cite{bkk10}, and hence strong convexity, holds. Then we may seek a minimizer $\varepsilon\in U^\varepsilon$ of $F$ by applying any gradient based method (such as steepest descent, quasi-Newton, or conjugate gradient), starting from $\varepsilon_0$.

Such an approach requires that we compute the Fr\'echet derivative of $F$, which is complicated since it involves the implicit dependence of $\mathbf{E}_\varepsilon$ upon $\varepsilon$. To simplify the analysis, in the spirit of optimal control (see for example \cite{bkr00, kl10} for the general theory and some specific examples), we introduce the Lagrangian
\begin{equation*} \label{lagrangian}
L(u)\coloneqq F(\varepsilon,\,\mathbf{E}) + \mathscr{D}(\varepsilon,\,\mathbf{E},\,\l),
\end{equation*}
where $u=(\varepsilon,\,\mathbf{E},\,\l)\in U \coloneqq U^\varepsilon\times V^\mathrm{dir}\times V^\mathrm{adj} \subset V\coloneqq V^\varepsilon\times V^\mathrm{dir}\times V^\mathrm{adj}$, $F(\varepsilon,\,\mathbf{E})$ was defined in \eqref{tikhonovfunctional}, and $\mathscr{D}(\varepsilon,\,\mathbf{E},\,\l)$ was defined in \eqref{weakmaxwell}.

We can now minimize $F$ over $U^\varepsilon$ by minimizing $L$ over $U$. With the strong convexity as above, this would imply that we solve
\begin{problem} \label{minlag}
Find $u\in U$ such that $L'(u;\,v)=0$ for every $v\in V$.
\end{problem}
Again we use the notation $L'(u;\,v)$ for the Fr\'echet derivative of $L$ at $u$, acting on $v$. It can be shown (see Proposition 1 of \cite{m14}) that
\begin{equation*}
L'(u;\,v)=\pd{L}{\varepsilon}(u;\,{\bar{\eps}}) + \pd{L}{\mathbf{E}}(u;\,{\bar{\E}}) + \pd{L}{\l}(u;\,{\bar{\l}}),
\end{equation*}
where $u=(\varepsilon,\,\mathbf{E},\,\l)\in U$, $v=({\bar{\eps}},\,{\bar{\E}},\,{\bar{\l}})\in V$, and
\begin{equation}
\begin{aligned} \label{dl}
\pd{L}{\varepsilon}(u;\,{\bar{\eps}}) &\coloneqq \alpha\sca{\varepsilon-\varepsilon_0}{{\bar{\eps}}}_\Omega
 - \sca{\pd{\mathbf{E}}{t}\cdot\pd{\l}{t}}{{\bar{\eps}}}_ {\Omega_T}
 + \sca{(\div\l)\mathbf{E}}{\nabla\left(\frac{{\bar{\eps}}}{\varepsilon}\right)}_ {\Omega_T} ,\\
 \pd{L}{\mathbf{E}}(u;\,{\bar{\E}}) &\coloneqq \sca{(\mathbf{E}-\mathbf{G})z_\delta^2}{{\bar{\E}}}_ {\Gamma_T}
 - \sca{\varepsilon\pd{\l}{t}}{\pd{{\bar{\E}}}{t}}_ {\Omega_T}
 + \sca{\nabla\l}{\nabla{\bar{\E}}}_ {\Omega_T} \\ &\qquad
 + \sca{\frac{\div\l}{\varepsilon}\nabla\varepsilon}{{\bar{\E}}}_ {\Omega_T}   \eqqcolon \mathscr{A}(\varepsilon,\,\l,\,{\bar{\E}}),\\
\pd{L}{\l}(u;\,{\bar{\l}}) &= \mathscr{D}(\varepsilon,\,\mathbf{E},\,{\bar{\l}}).
\end{aligned}
\end{equation}

In particular, we note that the solution $u=(\varepsilon,\,\mathbf{E},\,\l)$ to Problem~\ref{minlag} must satisfy $\mathscr{D}(\varepsilon,\,\mathbf{E},\,{\bar{\l}})=0$ for every ${\bar{\l}}\in V^\mathrm{adj}$ and $\mathscr{A}(\varepsilon,\,\l,\,{\bar{\E}}) = 0$ for every ${\bar{\E}}\in V^\mathrm{dir}$. The former means that $\mathbf{E}$ solves Problem~\ref{weakdirect} and the latter that $\l$ solves the following adjoint problem:
\begin{problem} \label{weakadjoint}
Given $\varepsilon\in U^\varepsilon$, determine $\l\in V^\mathrm{adj}$ such that $\mathscr{A}(\varepsilon,\,\l,\,\boldsymbol{\phi})=0$ for every $\boldsymbol{\phi}\in V^\mathrm{dir}$.
\end{problem}

The functional $\mathscr{A}$ in Problem~\ref{weakadjoint} was defined in \eqref{dl}. The problem can be seen as a weak analogue of the following system, adjoint to \eqref{maxwells}:
\begin{equation*}
\begin{aligned}
&\varepsilon\pd{^2\l}{t^2} - \Delta\l - \frac{\div\l}{\varepsilon}\nabla\varepsilon = 0 && \text{in }   {\Omega_T} ,\\
&\pd{\l}{\boldsymbol{\nu}}=-(\mathbf{E} - \mathbf{G})z_\delta^2 && \text{on }  {\Gamma_T} ,\\
&\l(\cdot,\,T) = \pd{\l}{t}(\cdot,\,T) = 0 && \text{in } \Omega.
\end{aligned}
\end{equation*}

These observations will be used in the error analysis to be described below. But first we shall make some remarks concerning the relation between the Fr\'echet derivative of Tikhonov functional and that of the Lagrangian.

Let $u_\varepsilon = (\varepsilon,\,\mathbf{E}_\varepsilon,\,\l_\varepsilon)$ be the element of $U$ obtained by taking $\mathbf{E}_\varepsilon$ as the solution to Problem~\ref{weakdirect} and $\l_\varepsilon$ as the solution to Problem~\ref{weakadjoint} for the given $\varepsilon\in U^\varepsilon$. Then, under assumption of sufficient stability of the weak solutions $\mathbf{E}_\varepsilon$ and $\l_\varepsilon$ with respect to $\varepsilon$, the observation that
\begin{equation*}
F(\varepsilon) = F(\varepsilon,\,\mathbf{E}_\varepsilon) = F(\varepsilon,\,\mathbf{E}_\varepsilon) + \mathscr{D}(\varepsilon,\,\mathbf{E}_\varepsilon,\,\l_\varepsilon) = L(u_\varepsilon),
\end{equation*}
(as $\mathscr{D}(\varepsilon,\,\mathbf{E}_\varepsilon,\,\l_\varepsilon)=0$) leads to
\begin{equation} \label{dj2dl}
F'(\varepsilon;\,\cdot) = \pd{L}{\varepsilon}(u_\varepsilon;\,\cdot).
\end{equation}
Estimate \eqref{convexity} and identity \eqref{dj2dl} will play an important role in the error analysis for the Tikhonov functional and for the coefficient.

\section{Finite element formulations and error analysis} \label{error_analysis}
In this section we will give finite element formulations for discretizing Problems~\ref{weakdirect}, \ref{minlag} and \ref{weakadjoint}. After that we will turn to the error analysis. We begin by defining finite-dimensional analogues of the spaces $V^\varepsilon$, $V^\mathrm{dir}$, $V^\mathrm{adj}$, and $V$, as well as subsets corresponding to $U^\varepsilon$ and $U$.

Let $\th\coloneqq\{K\}$ be a triangulation of $\Omega$ and let $\mathcal{I}_\tau$ be a uniform partition of $(0,\,T)$ into subintervals $(t_k,\,t_{k+1}]$, $t_k=k\tau$, $k=0,\,\ldots,\,N_\tau$, of length $\tau=T/N_\tau$. With $\th$ we associate a mesh-function $h=h(\mathbf{x})$ such that $h(\mathbf{x})=\operatorname{diam}(K)$ for $\mathbf{x}\in K\in \th$. On these meshes we define\footnote{Observe that the dependence on the step size $\tau$ in time is not explicitly included in the notation for the finite-dimensional spaces. This is justified by the fact that $\tau$ should be selected with regard to $h$ in accordance with the Courant-Friedrichs-Lewy condition.}
\begin{equation*}
\begin{aligned}
V_h^\varepsilon & \coloneqq \{v\in V^\varepsilon: v\rvert_K\in P^q(K)~\forall K\in \th\},\\
U_h^\varepsilon & \coloneqq V_h^\varepsilon\cap U^\varepsilon,\\
V_h}%{h,\,\tau}^\mathrm{dir}& \coloneqq \{v\in V^\mathrm{dir}:v\rvert_{K\times I}\in [P^1(K)]^3\times P^1(I)~\forall K\in\th~\forall I\in\mathcal{I}_\tau\},\\
V_h}%{h,\,\tau}^\mathrm{adj}& \coloneqq \{v\in V^\mathrm{adj}:v\rvert_{K\times I}\in [P^1(K)]^3\times P^1(I)~\forall K\in\th~\forall I\in\mathcal{I}_\tau\},\\
V_h}%{h,\,\tau} &\coloneqq V_h^\varepsilon \timesV_h}%{h,\,\tau}^\mathrm{dir}\timesV_h}%{h,\,\tau}^\mathrm{adj},\\
U_h}%{h,\,\tau} &\coloneqq U_h^\varepsilon \timesV_h}%{h,\,\tau}^\mathrm{dir}\timesV_h}%{h,\,\tau}^\mathrm{adj},
\end{aligned}
\end{equation*}
where $P^n(X)$ denotes the space of polynomials of degree at most $n\in\mathbb{N}$ over $X$, and the degree $q$ used in the finite-dimensional analogue $V_h^\varepsilon$ of $V^\varepsilon$ is at least 1.

Using these spaces we can state finite element versions of Problems~\ref{weakdirect} and \ref{weakadjoint} as Problem~\ref{femdirect} and Problem~\ref{femadjoint}, respectively, as follows:

\begin{problem}\label{femdirect}
Given $\varepsilon\in U^\varepsilon$, determine $\mathbf{E}_h\in V_h}%{h,\,\tau}^\mathrm{dir}$ such that $\mathscr{D}(\varepsilon,\,\mathbf{E}_h,\,\boldsymbol{\phi}_h)=0$ for every $\boldsymbol{\phi}_h\in V_h}%{h,\,\tau}^\mathrm{adj}$.
\end{problem}

\begin{problem}\label{femadjoint}
Given $\varepsilon\in U^\varepsilon$, determine $\l_h\in V_h}%{h,\,\tau}^\mathrm{adj}$ such that $\mathscr{A}(\varepsilon,\,\l_h,\,\boldsymbol{\phi}_h)=0$ for every $\boldsymbol{\phi}\in V_h}%{h,\,\tau}^\mathrm{dir}$.
\end{problem}

The finite-dimensional analogue for Problem~\ref{minlag} is:
\begin{problem} \label{femminlag}
Find $u_h=(\varepsilon_h,\,\mathbf{E}_h,\,\l_h)\in U_h}%{h,\,\tau}$ such that $L'(u_h,\,v)=0$ for every $v\in V_h}%{h,\,\tau}$.
\end{problem}
The same remark that was made in conjunction with Problem~\ref{minlag} is also valid here: it holds that $\mathbf{E}_h$ solves Problem~\ref{femdirect} and $\l_h$ solves Problem~\ref{femadjoint} for $\varepsilon=\varepsilon_h$.

We will now focus on estimations of the difference between the solution to Problem~\ref{minlag} and Problem~\ref{femminlag}. We begin by introducing some additional notation. For $v=(\varepsilon,\,\mathbf{E},\,\l)\in V$ we denote (with some slight abuse of notation) its interpolant in $V_h}%{h,\,\tau}$ by
\begin{align*}
{\Pi_h} v= ({\Pi_h}\varepsilon,\,{\Pi_h}\mathbf{E},\,{\Pi_h}\l),
\end{align*}
and the interpolation residual by
\begin{align*}
{r_h} v = v-{\Pi_h} v = ({r_h}\varepsilon,\, {r_h}\mathbf{E},\,{r_h}\l).
\end{align*}
 We will also need to consider jumps of discontinuous functions over $\th$ and $\mathcal{I}_\tau$. Let $K_1$, $K_2\in\th$ such that $\partial K_1\cap\partial K_2=e\neq\varnothing$. For $\mathbf{x}\in e$ we define
\begin{equation*}
\jm{v}{s}(\mathbf{x})\coloneqq\lim_{\mathbf{y}\to\mathbf{x},\,\mathbf{y}\in K_1}v(\mathbf{y}) + \lim_{\mathbf{y}\to\mathbf{x},\,\mathbf{y}\in K_2}v(\mathbf{y}),
\end{equation*}
so that in particular if $v=w\boldsymbol{\nu}$, where $w$ is piecewise constant on $\th$ and $\boldsymbol{\nu}$ is the outward unit normal, then $\jm{v}{s}=\jm{w\boldsymbol{\nu}}{s}=(w\boldsymbol{\nu})\rvert_{K_1} + (w\boldsymbol{\nu})\rvert_{K_2}$ is the normal jump across $e$. We extend $\jm{\cdot}{s}$ to every edge in $\th$ by defining $\jm{v}{s}(\mathbf{x})=0$ for $\mathbf{x}\in K\cap\Gamma$, $K\in \th$. The corresponding maximal jump is defined by
\begin{equation*}
\aj{v}{s}(\mathbf{x}) \coloneqq \max_{\mathbf{y}\in\partial K}\abs{\jm{v}{s}(\mathbf{y})}, \quad \mathbf{x}\in K\in \th.
\end{equation*}

For jumps in time, we define
\begin{equation*}
\jm{v}{t}(t_k)\coloneqq\begin{cases} {\displaystyle \lim_{s\to0+}}\big(v(t_k+s)-v(t_k-s)\big), & k=1,\,\ldots,\,N_\tau-1,\\
0 & k=0,\,N_\tau,\end{cases}
\end{equation*}
and
\begin{equation*}
\aj{v}{t}(t)\coloneqq
\max\{\abs{\jm{v}{t}(t_k)},\,\abs{\jm{v}{t}(t_{k+1})}\}
\quad t\in(t_k,\,t_{k+1}).
\end{equation*}

In the theorems and proofs to be presented, we will frequently use the symbols $\approx$ and $\lesssim$ to denote approximate equality and inequality, respectively, where higher order terms (with respect to mesh-size or errors) are neglected.

We are now ready to present the first a posteriori error estimate, an estimate for the Lagrangian. The theorem was first presented in \cite{m14}, but with only a very brief proof. We will here give the full details of the proof. Let us start by recalling the theorem:

\begin{theorem} (A posteriori error estimate for the Lagrangian.) \label{ape_lag}
Let $u=(\varepsilon,\,\mathbf{E},\,\l)\in U$ be the solution to Problem~\ref{minlag} and $u_h=(\varepsilon_h,\,\mathbf{E}_h,\,\l_h)\inU_h}%{h,\,\tau}$ be the solution to Problem~\ref{femminlag}. Then there exists a constant $C$, which does not depend on $u$, $u_h$, $h$, or $\tau$, such that
\begin{equation*}
\begin{aligned}
\abs{L(u) - L(u_h)} &\lesssim C\left( \sca{\abs{R_\varepsilon}}{h\abs{\aj{\pd{\varepsilon_h}{\boldsymbol{\nu}}}{s}}}_\Omega \right.\\
&\qquad + \left. \sca{R_{\l,\,\Omega}}{\tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}} + h\abs{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{s}}}_ {\Omega_T} \right.\\
&\qquad + \left. \sca{R_{\l,\,\Gamma}}{\tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}} + h\abs{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{s}}}_ {\Gamma_T} \right.\\
&\qquad + \left. \sca{R_{\mathbf{E},\,\Omega}}{\tau\abs{\aj{\pd{\l_h}{t}}{t}} + h\abs{\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}}}_ {\Omega_T} \right.\\
&\qquad + \left.\sca{R_{\mathbf{E},\,\Gamma}}{\tau\abs{\aj{\pd{\l_h}{t}}{t}} + h\abs{\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}}}_ {\Gamma_T} \right),
\end{aligned}
\end{equation*}
where
\begin{equation*}
\begin{aligned}
R_\varepsilon&=\alpha(\varepsilon_h - \varepsilon_0)
 - \int_0^T\pd{\mathbf{E}_h}{t}(\cdot,\,t)\cdot\pd{\l_h}{t}(\cdot,\,t)\,\mathrm{d} t\\&\quad
 - \int_0^T\frac{\div\mathbf{E}_h(\cdot,\,t)\div\l_h(\cdot,\,t)}{\varepsilon_h}\,\mathrm{d} t
 + \int_0^T\frac{\aj{(\boldsymbol{\nu}\cdot\mathbf{E}_h)(\div\l_h)}{s}}{h\varepsilon_h}\,\mathrm{d} t,\\
R_{\l,\,\Omega} &= -\varepsilon_h\frac{\aj{\pd{\l_h}{t}}{t}}{\tau}
 + \frac{\aj{\pd{\l}{\boldsymbol{\nu}}}{s}}{2h}
 + \frac{\div\l_h}{\varepsilon_h}\nabla\varepsilon_h,\\
R_{\l,\,\Gamma}&= \pd{\l_h}{\boldsymbol{\nu}} + (\mathbf{E}_h - \mathbf{G})z_\delta^2,\\
R_{\mathbf{E},\,\Omega}&= -\varepsilon_h\frac{\aj{\pd{\mathbf{E}_h}{t}}{t}}{\tau}
 + \frac{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{s}}{2h}
 + \frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h^2}\nabla\varepsilon_h
 - \frac{J_{\nabla\varepsilon_h}^\mathsf{T}\mathbf{E}_h + J_{\mathbf{E}_h}^\mathsf{T}\nabla\varepsilon_h}{\varepsilon_h} \\&\quad
 + \frac{\aj{(\nabla\varepsilon_h\cdot\mathbf{E}_h)\boldsymbol{\nu}}{s}}{2h\varepsilon_h},\\
R_{\mathbf{E},\,\Gamma}&= \pd{\mathbf{E}_h}{\boldsymbol{\nu}}-\P.
\end{aligned}
\end{equation*}
Here $J_{\nabla\varepsilon_h}$ and $J_{\mathbf{E}_h}$ denotes the Jacobi matrices of $\nabla\varepsilon_h$ and $\mathbf{E}_h$, respectively, and $(\cdot)^\mathsf{T}$ denotes matrix transpose.
\end{theorem}

Note that if $\varepsilon_h$ is piecewise linear, $J_{\nabla\varepsilon_h}\rvert_K\equiv0$ for every $K\in\th$, hence the corresponding term in $R_{\mathbf{E},\,\Omega}$ vanishes in that case.

In the following proof, and thereafter, $C$ is used to denote various constants of moderate size which do not depend on $u$, $u_h$, $h$, or $\tau$.

\begin{proof}
Using the definition of the Fr\'echet derivative we get
\begin{align*}
L(u) - L(u_h) &= L'(u_h;\,u - u_h) + o(\norm{u - u_h}_V)
\end{align*}
The split $u-u_h= (u - {\Pi_h} u) + ({\Pi_h} u - u_h) = {r_h} u + ({\Pi_h} u - u_h)$ now gives
\begin{align*}
L(u) - L(u_h) &= L'(u_h;\,{r_h} u + ({\Pi_h} u - u_h)) + o(\norm{u - u_h}_V)\\
 &= L'(u_h;\,{r_h} u) + L'(u_h;\, {\Pi_h} u - u_h) + o(\norm{u - u_h}_V).
\end{align*}
The second term vanishes since ${\Pi_h} u-u_h\in V_h$ and $u_h$ solves Problem~\ref{femminlag}, and we neglect the remainder term $o(\norm{u - u_h}_V)$ since it is of higher order with respect to the error. We are then left with
\begin{equation*}
L(u) - L(u_h) \approx L'(u_h;\,{r_h} u) = \pd{L}{\varepsilon}(u_h;\,{r_h}\varepsilon) + \pd{L}{\mathbf{E}}(u_h;\,{r_h}\mathbf{E}) + \pd{L}{\l}(u_h;\,{r_h}\l),
\end{equation*}
and individual estimation of these three terms will give the stated result.

Starting with the first term, we observe that
\begin{equation*}
\begin{aligned}
\pd{L}{\varepsilon}(u_h;\,{r_h}\varepsilon) =
\alpha\sca{\varepsilon_h - \varepsilon_0}{{r_h}\varepsilon}_\Omega
& -\sca{\pd{\mathbf{E}_h}{t}\cdot\pd{\l_h}{t}}{{r_h}\varepsilon}_ {\Omega_T}  \\
& + \sca{(\div\l_h)\mathbf{E}_h}{\nabla\left(\frac{{r_h}\varepsilon}{\varepsilon_h}\right)}_ {\Omega_T} .
\end{aligned}
\end{equation*}

We aim at lifting all derivatives from the interpolation residuals, thus we split the inner product over $ {\Omega_T} $ in the last term above into the sum of inner products over $K_T\coloneqq K\times(0,\,T)$, $K\in \th$:
\begin{equation*}
\sca{(\div\l_h)\mathbf{E}_h}{\nabla\left(\frac{{r_h}\varepsilon}{\varepsilon_h}\right)}_ {\Omega_T}  = \sum_{K\in\th}\sca{(\div\l_h)\mathbf{E}_h}{\nabla\left(\frac{{r_h}\varepsilon}{\varepsilon_h}\right)}_{K_T}.
\end{equation*}

We now integrate by parts, using the notation $\partial K_T\coloneqq \partial K\times(0,\,T)$, $\partial K'_T\coloneqq (\partial K\setminus \Gamma)\times(0,\,T)$, $\partial K''_T\coloneqq (\partial K\cap\Gamma)\times(0,\,T)$, $K\in\th$:
\begin{align*}
&\sum_{K\in\th}\sca{(\div\l_h)\mathbf{E}_h}{\nabla\left(\frac{{r_h}\varepsilon}{\varepsilon_h}\right)}_{K_T}\\
&\quad=\sum_{K\in\th}\left(-\sca{\div\big((\div\l_h)\mathbf{E}_h\big)}{\frac{{r_h}\varepsilon}{\varepsilon_h}}_{K_T}
 + \sca{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{\frac{{r_h}\varepsilon}{\varepsilon_h}}_{\partial K_T}\right)\\
&\quad = \sum_{K\in\th}\left(-\sca{\frac{\nabla(\div\l_h)\cdot\mathbf{E}_h}{\varepsilon_h}}{{r_h}\varepsilon}_{K_T}
 - \sca{\frac{(\div\l_h)(\div\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_{K_T}\right.\\
&\qquad\qquad\quad\left.+\sca{\frac{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_{\partial K_T'}
 + \sca{\frac{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_{\partial K_T''}\right)\\
&\quad = -\sum_{K\in\th}\sca{\frac{\nabla(\div\l_h)\cdot\mathbf{E}_h}{\varepsilon_h}}{{r_h}\varepsilon}_{K_T}
 - \sca{\frac{(\div\l_h)(\div\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_ {\Omega_T}  \\
&\qquad + \sum_{K\in\th}\sca{\frac{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_{\partial K_T'}
 + \sca{\frac{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_ {\Gamma_T} .
\end{align*}

We observe that $\nabla(\div\l_h)\equiv0$ on every $K_T$, $K\in\th$, since $\l_h$ is piecewise linear, and that $\varepsilon\equiv1$ on $\Gamma$ so that ${r_h}\varepsilon\rvert_\Gamma\equiv0$. With this in mind, the above calculations yields
\begin{align*}
\sca{(\div\l_h)\mathbf{E}_h}{\nabla\left(\frac{{r_h}\varepsilon}{\varepsilon_h}\right)}_ {\Omega_T}  &=
 - \sca{\frac{(\div\l_h)(\div\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_ {\Omega_T}  \\&\quad
 + \sum_{K\in\th}\sca{\frac{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_{\partial K_T'}.
\end{align*}

In order to obtain a residual defined in the whole of $\Omega$, as opposed to one containing terms defined only on edges of elements $K\in\th$, we should manipulate the last term in the above expression further. Observe that
\begin{align*}
\sum_{K\in\th}\sca{\frac{1}{\varepsilon_h}(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{{r_h}\varepsilon}_{\partial K_T'} = \frac{1}{2}\sum_{K\in\th}\sca{\frac{1}{\varepsilon_h}\jm{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{s}}{{r_h}\varepsilon}_{\partial K_T'},
\end{align*}
where the factor $\frac{1}{2}$ appears since every internal edge is counted exactly twice in the sum over all elements $K\in \th$.

Using the approximation
\begin{equation*}
\int_{\partial K}f\,\mathrm{d} S \approx \int_K\frac{\tilde{f}}{h_K}\,\mathrm{d}\mathbf{x}
\end{equation*}
where $\tilde{f}$ denotes the maximum of $f$ over $\partial K$ (see for instance \cite{eehj96}), we finally get
\begin{align*}
\sum_{K\in\th}\sca{\frac{1}{\varepsilon_h}\jm{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{s}}{{r_h}\varepsilon}_{\partial K_T'}&\approx
\sum_{K\in\th}\sca{\frac{1}{h_K\varepsilon_h}\aj{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{s}}{{r_h}\varepsilon}_{K_T}\\
&= \sca{\frac{1}{h\varepsilon_h}\aj{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{s}}{{r_h}\varepsilon}_ {\Omega_T} ,
\end{align*}
which gives
\begin{align*}
\pd{L}{\varepsilon}(u_h;\,{r_h}\varepsilon) &=
\alpha\sca{\varepsilon_h - \varepsilon_0}{{r_h}\varepsilon}_\Omega
 + \sca{\pd{\mathbf{E}_h}{t}\cdot\pd{\l_h}{t}}{{r_h}\varepsilon}_ {\Omega_T}  \\&\qquad
 - \sca{\frac{(\div\l_h)(\div\mathbf{E}_h)}{\varepsilon_h}}{{r_h}\varepsilon}_ {\Omega_T}
 + \sca{\frac{1}{h\varepsilon_h}\aj{(\div\l_h)(\boldsymbol{\nu}\cdot\mathbf{E}_h)}{s}}{{r_h}\varepsilon}_ {\Omega_T}  \\&
 = \sca{R_\varepsilon}{{r_h}\varepsilon}_\Omega.
\end{align*}

We can now estimate ${r_h}\varepsilon$ in terms of $\varepsilon_h$, using standard interpolation techniques (see for instance \cite{js95}), as
\begin{equation*}
\abs{{r_h}\varepsilon}\leq Ch^2\abs{D^2\varepsilon}\approx Ch^2\abs{\frac{\aj{\pd{\varepsilon_h}{\boldsymbol{\nu}}}{s}}{h}}=Ch\abs{\aj{\pd{\varepsilon_h}{\boldsymbol{\nu}}}{s}},
\end{equation*}
where $D^2$ denotes derivatives of second order with respect to $\mathbf{x}$. Thus
\begin{equation*}
\abs{\pd{L}{\varepsilon}(u_h;\,{r_h}\varepsilon)}\lesssim C\sca{\abs{R_\varepsilon}}{h\abs{\aj{\pd{\varepsilon_h}{\boldsymbol{\nu}}}{s}}}_\Omega.
\end{equation*}

We continue with
\begin{align*}
\pd{L}{\mathbf{E}}(u_h;\,{r_h}\mathbf{E}) &= \sca{(\mathbf{E}_h - \mathbf{G})z_\delta^2}{{r_h}\mathbf{E}}_ {\Gamma_T}
 - \sca{\varepsilon_h\pd{\l_h}{t}}{\pd{{r_h}\mathbf{E}}{t}}_ {\Omega_T}  \\&\qquad
 + \sca{\nabla\l_h}{\nabla{r_h}\mathbf{E}}_ {\Omega_T}
 + \sca{\frac{\div\l_h}{\varepsilon_h}\nabla\varepsilon_h}{{r_h}\mathbf{E}}_ {\Omega_T} .
\end{align*}
Again, we seek to lift derivatives from the interpolation residuals, and so we use integration by parts to get
\begin{align*}
\sca{\varepsilon_h\pd{\l_h}{t}}{\pd{{r_h}\mathbf{E}}{t}}_ {\Omega_T}  &= \sum_{k=1}^{N_\tau}\int_{t_{k-1}}^{t_k}\sca{\varepsilon_h\pd{\l_h}{t}}{\pd{{r_h}\mathbf{E}}{t}}_\Omega\,\mathrm{d} t \\
&= -\sum_{k=1}^{N_\tau}\int_{t_{k-1}}^{t_k}\sca{\varepsilon_h\pd{^2\l_h}{t^2}}{{r_h}\mathbf{E}}_\Omega\,\mathrm{d} t \\&\quad
 + \sum_{k=1}^{N_\tau}\Big(\sca{\varepsilon_h\pd{\l_h}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=t_k} - \sca{\varepsilon_h\pd{\l_h}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=t_{k-1}}\Big)\\
&= \sum_{k = 1}^{N_\tau-1}\sca{\varepsilon_h\jm{\pd{\l_h}{t}}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=t_k}
 + \sca{\varepsilon_h\pd{\l_h}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=T} \\
&\quad - \sca{\varepsilon_h\pd{\l_h}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=0},
\end{align*}
where we have used the fact that $\pd{^2\l_h}{t^2}\equiv0$ on each subinterval $(t_{k-1},\,t_k)$, for the piecewise linear function $\l_h$.

Since ${r_h}\mathbf{E}(\cdot,\,0)=\pd{\l_h}{t}(\cdot,\,T)=0$, this leaves us with
\begin{equation*}
\sca{\varepsilon_h\pd{\l_h}{t}}{\pd{{r_h}\mathbf{E}}{t}}_ {\Omega_T}  = \sum_{k = 1}^{N_\tau-1}\sca{\varepsilon_h\jm{\pd{\l_h}{t}}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=t_k}.
\end{equation*}
We now approximate the boundary terms by terms defined on the whole interval, using
\begin{equation*}
f(t_k)\approx \frac{1}{\tau}\int_{t_{k-1}}^{t_k}f(t)\,\mathrm{d} t,
\end{equation*}
that is
\begin{align*}
\sum_{k = 1}^{N_\tau-1}\sca{\varepsilon_h\jm{\pd{\l_h}{t}}{t}}{{r_h}\mathbf{E}}_\Omega\rvert_{t=t_k}&\approx\sum_{k = 1}^{N_\tau-1}\frac{1}{\tau}\int_{t_{k-1}}^{t_k}\sca{\varepsilon_h\aj{\pd{\l_h}{t}}{t}}{{r_h}\mathbf{E}}_\Omega\,\mathrm{d} t\\
&=\sca{\frac{\varepsilon_h}{\tau}\aj{\pd{\l_h}{t}}{t}}{{r_h}\mathbf{E}}_ {\Omega_T} .
\end{align*}

Moving on to
\begin{equation*}
\sca{\nabla\l_h}{\nabla{r_h}\mathbf{E}}_ {\Omega_T}  = \sum_{K\in\th}\sca{\nabla\l_h}{\nabla{r_h}\mathbf{E}}_{K_T},
\end{equation*}
we integrate by parts and use the fact that $\Delta\l_h\equiv0$ in every $K\in\th$ to obtain
\begin{align*}
\sca{\nabla\l_h}{\nabla{r_h}\mathbf{E}}_ {\Omega_T}  &=\sum_{K\in\th}\left(-\sca{\Delta\l_h}{{r_h}\mathbf{E}}_{K_T} + \sca{\pd{\l_h}{\boldsymbol{\nu}}}{{r_h}\mathbf{E}}_{\partial K_T}\right)\\
&=\sum_{K\in\th}\left(\sca{\pd{\l_h}{\boldsymbol{\nu}}}{{r_h}\mathbf{E}}_{\partial K'_T} + \sca{\pd{\l_h}{\boldsymbol{\nu}}}{{r_h}\mathbf{E}}_{\partial K''_T}\right)\\
&=\frac{1}{2}\sum_{K\in\th}\sca{\jm{\pd{\l_h}{\boldsymbol{\nu}}}{s}}{{r_h}\mathbf{E}}_{\partial K'_T} + \sca{\pd{\l_h}{\boldsymbol{\nu}}}{{r_h}\mathbf{E}}_ {\Gamma_T} .
\end{align*}

We again approximate inner products over $\partial K'_T$ by inner products over $K_T$, so that
\begin{equation*}
\sca{\nabla\l_h}{\nabla{r_h}\mathbf{E}}_ {\Omega_T}  \approx \sca{\frac{1}{2h}\aj{ \pd{\l_h}{\boldsymbol{\nu}}}{s}}{{r_h}\mathbf{E}}_ {\Omega_T}  + \sca{\pd{\l_h}{\boldsymbol{\nu}}}{{r_h}\mathbf{E}}_ {\Gamma_T} .
\end{equation*}
Together with previous calculations, this gives
\begin{align*}
\pd{L}{\mathbf{E}}(u_h;\,{r_h}\mathbf{E}) &\approx  \sca{(\mathbf{E}_h - \mathbf{G})z_\delta^2}{{r_h}\mathbf{E}}_ {\Gamma_T}
 - \sca{\frac{\varepsilon_h}{\tau}\aj{\pd{\l_h}{t}}{t}}{{r_h}\mathbf{E}}_ {\Omega_T}  \\&\qquad
 + \sca{\frac{1}{2h}\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}}{{r_h}\mathbf{E}}_ {\Omega_T}
 + \sca{\pd{\l_h}{\boldsymbol{\nu}}}{{r_h}\mathbf{E}}_ {\Gamma_T}  \\&\qquad
 + \sca{\frac{\div\l_h}{\varepsilon_h}\nabla\varepsilon_h}{{r_h}\mathbf{E}}_ {\Omega_T} \\
&=\sca{R_{\l,\,\Omega}}{{r_h}\mathbf{E}}_ {\Omega_T}  + \sca{R_{\l,\,\Gamma}}{{r_h}\mathbf{E}}_ {\Gamma_T} .
\end{align*}
We once more use interpolation estimates
\begin{equation*}
\abs{{r_h}\mathbf{E}}\leq C\left(h^2\abs{D^2\mathbf{E}} + \tau^2\abs{\pd{^2\mathbf{E}}{t^2}} \right)\approx C\left(h\abs{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{t}} + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{s}} \right).
\end{equation*}
to get
\begin{align*}
\abs{\pd{L}{\mathbf{E}}(u_h;\,{r_h}\mathbf{E})} &\lesssim C\left(\sca{\abs{R_{\l,\,\Omega}}}{h\abs{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{t}} + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{s}} }_ {\Omega_T} \right. \\&\qquad
 + \left. \sca{\abs{R_{\l,\,\Gamma}}}{h\abs{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{t}} + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{s}}}_ {\Gamma_T} \right).
\end{align*}

It remains to estimate
\begin{align*}
\pd{L}{\l}(u_h;\,{r_h}\l) &=
 - \sca{\varepsilon_h\pd{\mathbf{E}_h}{t}}{\pd{{r_h}\l}{t}}_ {\Omega_T}
 + \sca{\nabla\mathbf{E}_h}{\nabla{r_h}\l}_ {\Omega_T}  \\&\quad
 + \sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}}{\div{r_h}\l}_ {\Omega_T}
 - \sca{\P}{{r_h}\l}_ {\Gamma_T} .
\end{align*}
Just as before, we obtain
\begin{equation*}
\sca{\varepsilon_h\pd{\mathbf{E}_h}{t}}{\pd{{r_h}\l}{t}}_ {\Omega_T}  \approx \sca{\frac{\varepsilon_h}{\tau}\aj{\pd{\mathbf{E}_h}{t}}{t}}{{r_h}\l}_ {\Omega_T}
\end{equation*}
and
\begin{equation*}
\sca{\nabla\mathbf{E}_h}{\nabla{r_h}\l}_ {\Omega_T} \approx \sca{\frac{1}{2h} \aj{ \pd{\mathbf{E}_h}{\boldsymbol{\nu}} }{s} }{{r_h}\l}_ {\Omega_T}  + \sca{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{{r_h}\l}_ {\Gamma_T} .
\end{equation*}
Consider the term
\begin{equation*}
\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}}{\div{r_h}\l}_ {\Omega_T}  = \sum_{K\in\th}\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}}{\div{r_h}\l}_{K_T}.
\end{equation*}
Integration by parts yields
\begin{align*}
&\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}}{\div{r_h}\l}_ {\Omega_T}  \\&\qquad=
\sum_{K\in\th}\left(-\sca{\nabla\left(\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}\right)}{{r_h}\l}_{K_T}
  + \sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}\boldsymbol{\nu}}{{r_h}\l}_{\partial K_T}\right) \\&\qquad=
\sum_{K\in\th}\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h^2}\nabla\varepsilon_h - \frac{J^\mathsf{T}_{\nabla\varepsilon_h}\mathbf{E}_h + J^\mathsf{T}_{\mathbf{E}_h}\nabla\varepsilon_h}{\varepsilon_h} }{{r_h}\l}_{K_T}\\&\qquad\quad
 + \sum_{K\in\th}\left(\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}\boldsymbol{\nu}}{{r_h}\l}_{\partial K'_T}
 + \sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}\boldsymbol{\nu}}{{r_h}\l}_{\partial K''_T}\right) \\&\qquad=
\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h^2}\nabla\varepsilon_h - \frac{J^\mathsf{T}_{\nabla\varepsilon_h}\mathbf{E}_h + J^\mathsf{T}_{\mathbf{E}_h}\nabla\varepsilon_h}{\varepsilon_h} }{{r_h}\l}_ {\Omega_T}  \\&\qquad\quad
+ \sum_{K\in\th}\sca{\frac{1}{\varepsilon_h}\jm{(\nabla\varepsilon_h\cdot\mathbf{E}_h)\boldsymbol{\nu}}{s}}{{r_h}\l}_{\partial K'_T}
+ \sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}\boldsymbol{\nu}}{{r_h}\l}_ {\Gamma_T} ,
\end{align*}
where for the second equality we have used the identity $\nabla(\nabla\varepsilon_h\cdot\mathbf{E}_h)=J^\mathsf{T}_{\nabla\varepsilon_h}\mathbf{E}_h + J^\mathsf{T}_{\mathbf{E}_h}\nabla\varepsilon_h$.

Noting that $\nabla\varepsilon_h\rvert_\Gamma\equiv0$ as $\varepsilon_h\in U^\varepsilon_h$, and using the usual approximation for $\jm{\cdot}{s}$ inside elements $K\in\th$ we get
\begin{align*}
\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h}}{\div{r_h}\l}_ {\Omega_T}  &\approx
\sca{\frac{\nabla\varepsilon_h\cdot\mathbf{E}_h}{\varepsilon_h^2}\nabla\varepsilon_h - \frac{J^\mathsf{T}_{\nabla\varepsilon_h}\mathbf{E}_h + J^\mathsf{T}_{\mathbf{E}_h}\nabla\varepsilon_h}{\varepsilon_h}}{{r_h}\l}\\&\quad + \sca{\frac{1}{2h\varepsilon_h}\aj{(\nabla\varepsilon_h\cdot\mathbf{E}_h)\boldsymbol{\nu}}{s}}{{r_h}\l}_ {\Omega_T} .
\end{align*}

Combining the results for $\pd{L}{\l}(u_h;\,{r_h}\l)$ and estimating ${r_h}\l$ in terms of $\l_h$ just as ${r_h}\mathbf{E}$ was estimated in terms of $\mathbf{E}_h$ gives
\begin{align*}
\abs{\pd{L}{\l}(u_h;\,{r_h}\l)
&\lesssim C\left(\sca{\abs{R_{\mathbf{E},\,\Omega}}}{h\abs{\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}}  + \tau\abs{\aj{\pd{\l_h}{t}}{t}}}_ {\Omega_T}  \right.\\& \qquad + \left.\sca{\abs{R_{\mathbf{E},\,\Gamma}}}{h\abs{\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}} + \tau\abs{\aj{\pd{\l_h}{t}}{t}}}_ {\Gamma_T} \right),
\end{align*}
which completes the proof.
\end{proof}

One should note that the terms in the error estimate of Theorem~\ref{ape_lag} which are derived from $\pd{L}{\l}(u_h;\,{r_h}\l)$ and $\pd{L}{\mathbf{E}}(u_h;\,{r_h}\mathbf{E})$ estimate how accurately the solutions of Problem~\ref{weakdirect} and Problem~\ref{weakadjoint} are approximated by the solutions of Problem~\ref{femdirect} and Problem~\ref{femadjoint}, respectively, for the approximate coefficient $\varepsilon_h$. The remaining term, $\sca{R_\varepsilon}{h\abs{\aj{\pd{\varepsilon_h}{\boldsymbol{\nu}}}{s}}}_\Omega$ can be interpreted as the error induced by approximating $\varepsilon$ by $\varepsilon_h$. Thus, if we are mainly interested in that error, or if we can postulate that the finite element approximations $\mathbf{E}_h$ and $\l_h$ are computed with relatively high accuracy, then $\abs{R_\varepsilon}$ may be used as an error indicator by itself. The significance of $R_\varepsilon$ will be further illustrated by the error estimates for the coefficient and for the Tikhonov functional.

We now proceed with an error estimate for the coefficient itself. An error estimate for the Tikhonov functional will follow as a corollary.

\begin{theorem} (A posteriori error estimate for the coefficient.) \label{ape_norm}
Suppose that the initial approximation $\varepsilon_0$ and the regularization parameter $\alpha$ are such that the strong convexity estimate \eqref{convexity} holds. Let $u=(\varepsilon,\,\mathbf{E},\,\l)\in U$ be the solution to Problem~\ref{minlag}, and let $u_h=(\varepsilon_h,\,\mathbf{E}_h,\,\l_h)\in U_h}%{h,\,\tau}$ be the solution to Problem~\ref{femminlag}, computed on meshes $\th$ and $\mathcal{I}_\tau$. Denote by $\tilde\mathbf{E}$ and $\tilde\l$ the solutions to Problem~\ref{weakdirect} and Problem~\ref{weakadjoint}, respectively, with permittivity $\varepsilon_h$, and set $\tilde{u} =(\varepsilon_h,\,\tilde\mathbf{E},\,\tilde\l)\in U$. Then there exists a constant $C$, which does not depend on $u$, $u_h$, $h$, or $\tau$, such that
\begin{equation*}
\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon} \lesssim C( c_\varepsilon\eta + \norm{R_\varepsilon}_\Omega),
\end{equation*}
where $c_\varepsilon\coloneqq\max\{1,\,\norm{\nabla\varepsilon_h}_{L_\infty(\Omega)}\}$ and $\eta =  \eta(u_h)$ is defined by
\begin{align*}
\eta &\coloneqq
   \sca{ \frac{1}{\tau} \abs{\aj{\pd{\l_h}{t}}{t}} + \abs{\div\l_h} }{h\abs{\aj{\pd{\mathbf{E}_h}{\boldsymbol{\nu}}}{s}} + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}}  }_ {\Omega_T}  \\ &\qquad
 + \sca{\frac{1}{\tau}\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}}}{h\abs{\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}} + \tau\abs{\aj{\pd{\l_h}{t}}{t}} }_ {\Omega_T} \\&\qquad
 + \sca{\abs{\mathbf{E}_h}}{ \abs{\aj{\pd{\l_h}{\boldsymbol{\nu}}}{s}} + \tau\abs{\aj{\pd{\div\l_h}{t}}{t}} }_ {\Omega_T} .
\end{align*}
\end{theorem}

\begin{proof}
Using strong convexity \eqref{convexity}, we obtain
\begin{equation*}
\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}^2\leq c \left(F'(\varepsilon;\,\varepsilon - \varepsilon_h) - F'(\varepsilon_h;\,\varepsilon-\varepsilon_h)\right).
\end{equation*}
Since $\varepsilon$ minimizes $F(\varepsilon)$ we have $F'(\varepsilon;\varepsilon-\varepsilon_h)=0$ and thus
\begin{equation} \label{pfthm2_00}
\norm{\varepsilon-\varepsilon_h}_{V^\varepsilon}^2\leq c\abs{F'(\varepsilon_h;\,\varepsilon - \varepsilon_h)} = c\abs{\pd{L}{\varepsilon}(\tilde u;\,\varepsilon - \varepsilon_h)},
\end{equation}
where the last equality follows from \eqref{dj2dl}.

We expand
\begin{equation} \label{pfthm2_0}
\begin{aligned}
\abs{\pd{L}{\varepsilon}(\tilde u;\,\varepsilon - \varepsilon_h)} &= \abs{\pd{L}{\varepsilon}(\tilde u;\,\varepsilon - \varepsilon_h) -\pd{L}{\varepsilon}(u_h;\,\varepsilon - \varepsilon_h) + \pd{L}{\varepsilon}(u_h;\,\varepsilon - \varepsilon_h)}\\
&\leq \abs{\pd{L}{\varepsilon}(\tilde u;\,\varepsilon - \varepsilon_h) -\pd{L}{\varepsilon}(u_h;\,\varepsilon - \varepsilon_h)} + \abs{\pd{L}{\varepsilon}(u_h;\,\varepsilon - \varepsilon_h)}\\
&\eqqcolon \abs{\Theta_1} + \abs{\Theta_2},
\end{aligned}
\end{equation}
and estimate the two terms $\abs{\Theta_1}$ and $\abs{\Theta_2}$ separately.

For $\Theta_1$ we use the linearization
\begin{align*}
\Theta_1&=
 \pd{^2L}{\mathbf{E}\partial\varepsilon}(u_h;\,\tilde\mathbf{E} - \mathbf{E}_h;\varepsilon - \varepsilon_h) + o(\lVert\tilde\mathbf{E} - \mathbf{E}_h\rVert_{ {\Omega_T} ,\,1}) \\ &\quad
 + \pd{^2L}{\l\partial\varepsilon}(u_h;\,\tilde\l - \l_h;\varepsilon - \varepsilon_h) + o(\lVert\tilde\l - \l_h\rVert_{ {\Omega_T} ,\,1}),
\end{align*}
where $\pd{^2L}{\mathbf{E}\partial\varepsilon}$ and $\pd{^2L}{\l\partial\varepsilon}$ denote mixed second partial Fr\'echet derivatives of $L$. Again, the remainder terms are neglected as they are of higher order with respect to the error. Thus, after exchanging the order of differentiation, we are left with
\begin{equation}
\begin{aligned}\label{pfthm2_1}
\Theta_1 &\approx D_1\rvert_{\varepsilon - \varepsilon_h}\left(\pd{L}{\mathbf{E}}(u_h;\,\tilde\mathbf{E} - \mathbf{E}_h) + \pd{L}{\l}(u_h;\,\tilde\l - \l_h)\right),
\end{aligned}
\end{equation}
where $D_1\rvert_{\varepsilon - \varepsilon_h}$ denotes differentiation with respect to the first component in $u_h$ and action on $\varepsilon - \varepsilon_h$.

We split $\tilde\mathbf{E} - \mathbf{E}_h=(\tilde\mathbf{E} - {\Pi_h}\tilde\mathbf{E}) + ({\Pi_h}\tilde\mathbf{E} - \mathbf{E}_h) = {r_h}\tilde\mathbf{E} + ({\Pi_h}\tilde\mathbf{E} - \mathbf{E}_h)$ and use the fact that $\l_h$ solves Problem~\ref{weakadjoint} with coefficient $\varepsilon_h$, so that $\pd{L}{\mathbf{E}}(u_h;\,{\Pi_h}\tilde\mathbf{E}-\mathbf{E}_h) = 0$ as ${\Pi_h}\tilde\mathbf{E}-\mathbf{E}_h\in V_h^\mathrm{dir}$. This gives
\begin{align} \label{pfthm2_2}
\pd{L}{\mathbf{E}}(u_h;\,\tilde\mathbf{E} - \mathbf{E}_h) = \pd{L}{\mathbf{E}}(u_h;\,{r_h}\tilde\mathbf{E}) + \pd{L}{\mathbf{E}}(u_h;\,{\Pi_h}\tilde\mathbf{E} - \mathbf{E}_h) = \pd{L}{\mathbf{E}}(u_h;\,{r_h}\tilde\mathbf{E}).
\end{align}
Similarly, we have
\begin{align} \label{pfthm2_3}
\pd{L}{\l}(u_h;\,\tilde\l - \l_h) = \pd{L}{\l}(u_h;\,{r_h}\tilde\l) + \pd{L}{\l}(u_h;\,{\Pi_h}\tilde\l - \l_h) = \pd{L}{\l}(u_h;\,{r_h}\tilde\l).
\end{align}
as $\mathbf{E}_h$ solves Problem~\ref{weakdirect} with coefficient $\varepsilon_h$.

Combining \eqref{pfthm2_1}, \eqref{pfthm2_2}, and \eqref{pfthm2_3} gives
\begin{align*}
\Theta_1 &\approx D_1\rvert_{\varepsilon - \varepsilon_h}\left(\pd{L}{\mathbf{E}}(u_h;\,{r_h}\tilde\mathbf{E}) + \pd{L}{\l}(u_h;\,{r_h}\tilde\l)\right)\\
 &= -\sca{(\varepsilon - \varepsilon_h)\pd{{r_h}\tilde\mathbf{E}}{t}}{\pd{\l_h}{t}}_ {\Omega_T}  + \sca{\nabla\left(\frac{\varepsilon - \varepsilon_h}{\varepsilon_h}\right){r_h}\tilde\mathbf{E}}{\div\l_h}_ {\Omega_T}  \\
 & \quad-\sca{(\varepsilon - \varepsilon_h)\pd{\mathbf{E}_h}{t}}{\pd{{r_h}\tilde\l}{t}}_ {\Omega_T}  + \sca{\nabla\left(\frac{\varepsilon - \varepsilon_h}{\varepsilon_h}\right)\mathbf{E}_h}{\div{r_h}\tilde\l}_ {\Omega_T} .
\end{align*}

In the same manner as in the proof of Theorem~\ref{ape_lag}, we integrate by parts in time and approximate jumps to get
\begin{align*}
-\sca{(\varepsilon - \varepsilon_h)\pd{{r_h}\tilde\mathbf{E}}{t}}{\pd{\l_h}{t}}_ {\Omega_T}  &= \sum_{k=1}^{N_\tau}\int_{t_{k-1}}^{t_k}\sca{(\varepsilon - \varepsilon_h){r_h}\tilde\mathbf{E}}{\jm{\pd{\l_h}{t}}{t}}_\Omega\,\mathrm{d} t \\
&\approx \sca{(\varepsilon - \varepsilon_h){r_h}\tilde\mathbf{E}}{\frac{1}{\tau}\aj{\pd{\l_h}{t}}{t}}_ {\Omega_T}
\end{align*}
and
\begin{align*}
-\sca{(\varepsilon - \varepsilon_h)\pd{\mathbf{E}_h}{t}}{\pd{{r_h}\tilde\l}{t}}_ {\Omega_T}  &= \sum_{k=1}^{N_\tau}\int_{t_{k-1}}^{t_k}\sca{ (\varepsilon - \varepsilon_h)\jm{ \pd{\mathbf{E}_h}{t} }{t} }{ {r_h}\tilde\l }_\Omega\,\mathrm{d} t\\
&\approx \sca{ (\varepsilon - \varepsilon_h)\frac{1}{\tau}\aj{ \pd{\mathbf{E}_h}{t} }{t} }{ {r_h}\tilde\l }_ {\Omega_T} .
\end{align*}
Thus
\begin{align*}
  \Theta_1 &\lesssim \sca{\abs{\varepsilon - \varepsilon_h}\frac{1}{\tau}\abs{\aj{\pd{\l_h}{t}}{t}}}{ \abs{{r_h}\tilde\mathbf{E}}}_ {\Omega_T}
 + \sca{\abs{\nabla\left(\frac{\varepsilon - \varepsilon_h}{\varepsilon_h}\right)}\abs{\div\l_h} }{ \abs{{r_h}\tilde\mathbf{E}}}_ {\Omega_T} \\
 & \quad+\sca{\abs{\varepsilon - \varepsilon_h}\frac{1}{\tau}\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}}}{\abs{{r_h}\tilde\l}}_ {\Omega_T}
 + \sca{\abs{\nabla\left(\frac{\varepsilon - \varepsilon_h}{\varepsilon_h}\right)}\abs{\mathbf{E}_h}}{\abs{\div{r_h}\tilde\l}}_ {\Omega_T} \\
&\leq \norm{\varepsilon - \varepsilon_h}_{L_\infty(\Omega)}\left(\sca{\frac{1}{\tau}\abs{\aj{\pd{\l_h}{t}}{t}}}{ \abs{{r_h}\tilde\mathbf{E}}}_ {\Omega_T}  + \sca{\frac{1}{\tau}\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}}}{\abs{{r_h}\tilde\l}}_ {\Omega_T} \right) \\&\quad
+ \norm{\nabla\left(\frac{\varepsilon - \varepsilon_h}{\varepsilon_h}\right)}_{L_\infty(\Omega)}\left(\sca{\abs{\div\l_h} }{ \abs{{r_h}\tilde\mathbf{E}}}_ {\Omega_T}  + \sca{\abs{\mathbf{E}_h}}{\abs{\div{r_h}\tilde\l}}_ {\Omega_T} \right).
\end{align*}

Note that
\begin{align*}
\norm{\nabla\left(\frac{\varepsilon - \varepsilon_h}{\varepsilon_h}\right)}_{L_\infty(\Omega)} &=
 \norm{\frac{\nabla(\varepsilon-\varepsilon_h)}{\varepsilon_h}  - \frac{(\varepsilon-\varepsilon_h)\nabla\varepsilon_h}{\varepsilon_h^2}}_{L_\infty(\Omega)} \\
&\leq\norm{\frac{1}{\varepsilon_h}}_{L_\infty(\Omega)}\norm{\nabla(\varepsilon - \varepsilon_h)}_{L_\infty(\Omega)} \\&\quad + \norm{\frac{1}{\varepsilon_h^2}}_{L_\infty(\Omega)}\norm{\nabla\varepsilon_h}_{L_\infty(\Omega)}\norm{\varepsilon-\varepsilon_h}_{L_\infty(\Omega)}
\end{align*}
and observe following facts:
\begin{align*}
\norm{\varepsilon-\varepsilon_h}_{L_\infty(\Omega)} + \norm{\nabla(\varepsilon-\varepsilon_h)}_{L_\infty(\Omega)} &\leq C\norm{\varepsilon-\varepsilon_h}_{V^\varepsilon},\\
\norm{\frac{1}{\varepsilon_h^p} }_{L_\infty(\Omega)} &\leq 1, \quad p \geq 0,
\end{align*}
the first following from the Sobolev inequality and the second from noting that $1\leq\varepsilon_h(\mathbf{x})\leq \varepsilon_\mathrm{max}$, $\mathbf{x}\in\Omega$, by \eqref{admissible}.

Using these observations, and interpolation estimates
\begin{align*}
\abs{{r_h}\tilde\mathbf{E}} &\leq C\left(h\abs{ \aj{ \pd{\mathbf{E}_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}}\right),\\
\abs{{r_h}\tilde\l} &\leq C\left(h\abs{ \aj{ \pd{\l_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\l_h}{t}}{t}}\right),\\
\abs{\div{r_h}\tilde\l} &\leq C\left(\abs{ \aj{ \pd{\l_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\div\l_h}{t}}{t}}\right),
\end{align*}
we get
\begin{equation}\label{pfthm2_4}
\begin{aligned}
 \Theta_ 1 &\lesssim C\left(\sca{\frac{1}{\tau}\abs{\aj{\pd{\l_h}{t}}{t}}}{h\abs{ \aj{ \pd{\mathbf{E}_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}} }_ {\Omega_T} \right. \\&\qquad
 + \left. \norm{\nabla\varepsilon_h}_{L_\infty(\Omega)}\sca{\abs{\div\l_h} }{ h\abs{ \aj{ \pd{\mathbf{E}_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}} }_ {\Omega_T}  \right.\\&\qquad
 + \left. \sca{\frac{1}{\tau}\abs{\aj{\pd{\mathbf{E}_h}{t}}{t}}}{ h\abs{ \aj{ \pd{\l_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\l_h}{t}}{t}} }_ {\Omega_T} \right.\\&\qquad + \left. \norm{\nabla\varepsilon_h}_{L_\infty(\Omega)}\sca{\abs{\mathbf{E}_h}}{\abs{ \aj{ \pd{\l_h}{\boldsymbol{\nu}} }{ s } } + \tau\abs{\aj{\pd{\div\l_h}{t}}{t}}}_ {\Omega_T}  \right)\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon} \\&
\leq Cc_\varepsilon\eta\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon},
\end{aligned}
\end{equation}
where $c_\varepsilon$ and $\eta$ were defined in the statement of the theorem.

Turning to $\Theta_2$ of \eqref{pfthm2_0}, we use the techniques of the proof of Theorem~\ref{ape_lag} to estimate
\begin{equation} \label{pfthm2_5}
\abs{\Theta_2} \lesssim C\sca{\abs{R_\varepsilon}}{\abs{\varepsilon-\varepsilon_h}}_\Omega \leq C\norm{R_\varepsilon}_\Omega\norm{\varepsilon - \varepsilon_h}_\Omega \leq C\norm{R_\varepsilon}_\Omega\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}.
\end{equation}

Combining estimates \eqref{pfthm2_4} and \eqref{pfthm2_5} with \eqref{pfthm2_00} and \eqref{pfthm2_0}, we conclude that
\begin{equation*}
\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}^2 \lesssim C\left(c_\varepsilon\eta\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon} + \norm{R_\varepsilon}_\Omega\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}\right),
\end{equation*}
and the result follows.
\end{proof}

Again, just as for the error estimate for the Lagrangian, we see that if the numerical errors for solving the direct and adjoint problems are relatively small, that is, when $\tilde u\approx u_h$ with relatively high accuracy, then $\norm{R_\varepsilon}_\Omega$ dominates the error estimate.

\begin{corollary}(A posteriori error estimate for the Tikhonov functional.) \label{ape_tikh}
Under the hypothesis of Theorem~\ref{ape_norm}, we have
\begin{equation*}
\abs{F(\varepsilon) - F(\varepsilon_h)} \lesssim C\left(c_\varepsilon^2\eta^2 + \norm{R_\varepsilon}_\Omega^2\right),
\end{equation*}
with $c_\varepsilon$ and $\eta$ as defined in Theorem~\ref{ape_norm}.
\end{corollary}

\begin{proof}
Using the definition of the Fr\'echet derivative and \eqref{dj2dl} we get
\begin{align*}
F(\varepsilon) - F(\varepsilon_h) &= F'(\varepsilon_h;\varepsilon - \varepsilon_h) + o(\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon})\\
                   &= \pd{L}{\varepsilon}(\tilde u;\,\varepsilon - \varepsilon_h) + o(\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}).
\end{align*}
Neglecting the remainder term as it is of higher order with respect to the error, and estimating $\pd{L}{\varepsilon}(\tilde u;\,\varepsilon - \varepsilon_h)$ as in the proof of Theorem~\ref{ape_norm}, we obtain
\begin{align*}
\abs{F(\varepsilon) - F(\varepsilon_h)} \lesssim C(c_\varepsilon\eta + \norm{R_\varepsilon}_{V^\varepsilon})\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}.
\end{align*}
Applying Theorem~\ref{ape_norm} to estimate $\norm{\varepsilon - \varepsilon_h}_{V^\varepsilon}$, we arrive at
\begin{align*}
\abs{F(\varepsilon) - F(\varepsilon_h)} \lesssim C(c_\varepsilon\eta + \norm{R_\varepsilon}_{V^\varepsilon})^2\leq C\left(c_\varepsilon^2\eta^2 + \norm{R_\varepsilon}_{V^\varepsilon}^2\right).
\end{align*}
\end{proof}

\section{Conclusion} \label{conclusion}
We have presented three a posteriori error estimates for an adaptive finite element method for the coefficient inverse problem, Problem \ref{cip}: for the Lagrangian, for the Tikhonov functional and for the coefficient. The latter two are presented here for the first time. Each estimator consists essentially of three parts, an estimate for the error resulting from finite element approximation of the solution to the direct problem, a similar estimate for the finite element approximation of the adjoint problem and an estimate corresponding to the approximation of the coefficient. The latter part is characterized by the residual $R_\varepsilon$ in all estimates.

Explicit solution schemes and numerical testing, including the proper choice of regularization parameter $\alpha$, will be the subject of forthcoming papers.

\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


text:
Long-Term Illness Scheme.
Paul Nicholas Gogarty
256 Mr. Gogarty asked the Tánaiste and Minister for Health and Children if there are plans to include irritable bowel syndrome as an illness under the Long Term and Disability Scheme Health Act 1970, in view of the increasing number of children and teenagers who have been diagnosed with Crohn's disease or ulcerative colitis. [25058/04]
Written answers (Question to Minister for Health)
Tánaiste and Minister for Health and Children (Ms Harney)
Under the 1970 Health Act, a health board may arrange for the supply, without charge, of drugs, medicines and medical and surgical appliances to people with a specified condition, for the treatment of that condition under the long-term illness scheme. The conditions are: mental handicap, mental illness, for people under 16 only, phenylketonuria, cystic fibrosis, spina bifida, hydrocephalus, diabetes mellitus, diabetes insipidus, haemophilia, cerebral palsy, epilepsy, multiple sclerosis, muscular dystrophies, parkinsonism, conditions arising from thalidomide and acute leukaemia. There are currently no plans to amend the list of eligible conditions.
Other schemes provide assistance towards the cost of approved drugs and medicines for people with significant ongoing medical expenses. People who cannot, without undue hardship, arrange for the provision of medical services for themselves and their dependants may be entitled to a medical card. Eligibility for a medical card is solely a matter for the chief executive officer of the relevant health board. In determining eligibility, the CEO has regard to the applicant's financial circumstances. Health boards use income guidelines to assist in determining eligibility. However, where a person's income exceeds the guidelines, a medical card may be awarded if the CEO considers that the person's medical needs or other circumstances would justify this. Medical cards may also be issued to individual family members on this basis. Non-medical card holders, and people with conditions not covered under the LTI, can use the drugs payment scheme. Under this scheme, no individual or family unit pays more than €78 per calendar month towards the cost of approved prescribed medicines.

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Berry, R. J. and H. N. Southern, editors.
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Mammal photography and observation: a practical field guide.
London: Academic Press, 1978. Octavo, photographs, line drawing, fine copy in dustwrapper.	More about Mammal photography and observation: a practical field guide.
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Measuring and monitoring biological diversity: standard methods for mammals.
Wilson, D. E. et al., editors.
Washington: Smithsonian Institution, 1996. Large octavo, paperback, illustrations.	More about Measuring and monitoring biological diversity: standard methods for mammals.
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Овсянка (также Большая Овсянка) — деревня в городском округе Семёновский Нижегородской области России. С 1845 года по 1930-е годы Овсянка была селом.

География
Деревня находится на севере центральной части Нижегородской области, в зоне хвойно-широколиственных лесов, на правом берегу реки Люнды, к югу от автодороги , на расстоянии приблизительно 30 километров (по прямой) к северо-востоку от города Семёнова, административного центра района. Абсолютная высота — 129 метров над уровнем моря. Через Овсянку проходит 45-й меридиан восточной долготы, который является эквивалентной долготой для часового пояса UTC+3:00 (или для Московского времени).

Климат
Климат характеризуется как умеренно континентальный, с влажным нежарким летом и холодной снежной зимой. Средняя температура воздуха самого холодного месяца (января) составляет −12,9 °C; самого тёплого месяца (июля) — 18,4 °C. Годовое количество атмосферных осадков — 550—600 мм.

Часовой пояс

Население

Национальный и религиозный состав
Согласно результатам переписи 2002 года, в национальной структуре населения русские составляли 98 %. По религиозному составу подавляющее большинство жителей - православные. В Овсянке с 2020 года возрождается храм Покрова Пресвятой Богородицы. В каждую пятницу в храме совершаются молебны, которые совершает настоятель отец Артемий Скакалин.

Примечания

Населённые пункты городского округа Семёновский


You can also make "tktktk" sounds or "tsss" sounds. Practice, practice, practice, so that you can make each of the above sounds successfully. Practice breathing on the beat, or whenever it feels comfortable in the music, but remember to breathe.... Make sure your mouth is closed and use the air inside of your lungs to force it through your nose. Now control that air by making by only letting a small amount of it out at a time but make sure the air that comes out moves quickly and sharply.
Standard Beatbox Notation (SBN) is a special set of musical notations created specifically for writing beat box songs. Хобби и стиль Комментарии • 1 593... How often do you find yourself in need of a beat but without a stereo? Don't get stuck without a beat, learn how to beatbox. Check out this video to learn how to make a "wobble wob wob" sound just like famous beatboxer Reeps One.
Hi! I'm Grant Zabriskie. I'm a 20-year beatboxer veteran with a passion for teaching and enabling young people to find and reach their dreams.
The Science Behind 'Beatboxing' beat-top-slider.jpg Many of the same mechanisms observed in human speech production were exploited for musical effect, including patterns of articulation that do not occur in the phonologies of the artist's native languages: ejectives, clicks and implosives.


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Zackawa (biał. Зацкава) – jezioro na Białorusi w rejonie grodzieńskim, położone ok. 26 km na północny wschód od Grodna w dorzeczu rzeki Pyranki.

Zackawa ma powierzchnię 0,75 km². Długość wynosi 2,1 km, zaś największa szerokość – 0,5 km. Długość linii brzegowej to 5,1 km. Brzegi porośnięte są lasem. Na południu łączy się z Jeziorem Białym. Na północy do Zackawy wpada rzeczka Szlamianka, łącząca z jeziorem Ant.

Bibliografia

 Беларуская энцыклапедыя: У 18 т. Т.7: Застаўка — Кантата / Рэдкал.: Г.П. Пашкоў і інш. — Мн.: Белэн, 1999. — 608 с.: іл.

Jeziora w obwodzie grodzieńskim


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You are here: Home / Games / Banjo Kazooie in Super Smash Bros Ultimate! Nintendo surprises at E3 2019
Banjo Kazooie in Super Smash Bros Ultimate! Nintendo surprises at E3 2019
One of Nintendo's surprises at E3 2019. The fans had long been asking for the chance to fight with Banjo & amp; Kazooie in the Smash Bros series, and the Great N has made his wishes come true. The sympathetic duo created by Rare in the days of Nintendo 64 will make the leap to the action of Super Smash Bros. Ultimate later this year, as we have seen in his most recent Nintendo Direct.
Other additions to the Nintendo crossover fight title, Banjo and his loyal companion Kazooie will feature combat movements faithful to the longed-for RARE platform, with a scenario inspired by the world of Banjo. Gruntilda, the Jinjos and other characters in this colorful universe will not miss their appointment with this downloadable content that will be released in the fall of 2019. The cinematic trailer has also surprised by its great sense of humor, playing with deception.
Recommended: Square Enix offers free Heavensward, expansion of Final Fantasy XIV.
Filed Under: Games, News Tagged With: Banjo & amp; Kazooie, Banjo & Kazooie, E3 2019, scenario inspired by the world of Banjo, Super Smash Bros. Ultimate


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layout: post
title: Proyek X Window
author: yuga
category: GNU/Linux
tags: [history, X]
---

## Proyek X Window

X Window System (X,X11,X Window) adalah evolusi graphical windowing system dari W Window. Pada awalnya W Window dikembangkan oleh Paul Asente dan Brian Reid dari Universitas Stanford untuk Sistem Operasi V. Tahun 1983, Paul Asente dan Chris Kent dengan mem-porting sistem W ke UNIX pada VS100 (VAXstation) untuk memberikan salinan kepada mereka yang bekerja di Laboratorium MIT.

Kemudian proyek X Window System pertama kali dimulai pada tahun 1984 yang merupakan  proyek open source,cross platform,dan software client-server.  Awalnya X dikembangkan di MIT sebagai bagian dari proyek Athena dan kerjasama antara MIT dengan DEC untuk mengembangkan rekayasa jaringan terminal heterogen yang dapat digunakan untuk pengajaran. Pada Mei 1984, X mulai dibuat oleh Robert Scheifler dari MIT dengan menggantikan protokol synchronous dari W dengan protokol asynchronous dan menghasilkan window system yang bernama 'X'.

19 Juni 1984, X  diumumkan kepada dunia melalui sebuah e-mail dari Robert Scheifler yang berisi:

*"Saya telah menghabiskan waktu beberapa minggu terakhir menulis sebuah sistem window untuk VS100. Saya mengambil cukup banyak kode dari W Window System yang di kelilingi dengan asynchronous  daripada antarmuka synchronous dan disebut sebagai X."*

15 September 1987, X11R1 dirilis yang merupakan rilis pertama dari X Window System. X11R1 merupakan desain ulang utama dan peningkatan X dari komunitas riset ke dalam rekayasa produk dan pengembangan masyarakat. X11R1 dimaksudkan untuk dapat mendukung semua hardware yang sudah ada dan hardware yang mendatang.

11 Februari 1991, munculah X386 yang ditulis oleh Thomas Roell dan merupakan implementasi pertama dari X untuk komputer yang kompatibel dengan IBM PC. Rilis pertama X386 adalah X386 1.1 yang berdasarkan X11R4.

1992, ketika David Wexelblat , Glenn Lai , David Dawes dan Jim Tsillas bergabung untuk mengatasi bug dalam kode sumber dari tampilan Server X386 ( ditulis oleh Thomas Roell ) , sebagai kontribusi untuk X11R5 dan hasilnya adalah X386 1.2E yang penyempurnaan versi X386 1.2.

Pengembangan X386 masih berlanjut dan didanai oleh SGCS (Snitily Graphics Consulting Services) dan versi rilisnya ditingkatkan secara komersial. SGCS kemudian menjadi Xi Graphics dan kode X386 menjadi close source kemudian berubah menjadi produk Accelerated-X.

redpajama_set_name: RedPajamaGithub


Did you know that it is possible to resolve many of the issues that you may be having with people if you are able to understand psychology? Once you understand how people think, you can determine what it is that is causing the problem from your perspective, and also from their perspective as well. The way that a person changes over the course of time as a result of the experiences that they have, and also the environment that they are in, is what developmental psychology is all about.
A quick definition is that it is the psychology of change, what causes people to shift from being one type of person to another, all because of how they are processing the world around them. To understand this, we need to look at what is developmental psychology, and how it is used to help people personally and professionally.
This is a type of psychology that is a branch of psychology that is used by many others. When you are dealing with children, adults, or even teenagers in high school, you need to know what is causing them to develop these problems. For example, you could have coworkers that are not able to get along. It may not be obvious why they are having problems, but a psychologist that understands how these changes occur can make modifications to the way that they process these interactions.
They can give them guidelines to follow and ultimately help resolve issues that may have been going on for years because of the strategies that they know. Developmental psychology is simply the understanding of how a person can take their situation and process it in a completely new way. That being said, let's look at this branch of psychology a little deeper to fully appreciate how powerful it actually is.
This is regarded as a scientific approach because it only looks at what has been tested and utilizes what has worked. It focuses on both adults and children, and how they change over time. Many of the theories that you will read about were developed by developmental psychologist that studied many areas. They would look at biology, emotions, cognitive processes and how a social environment could change the way a person processes information. It is dominated by Western thought, and as a result of this, it may not easily interact with those that use Eastern philosophies. However, it is very structured, and based upon this structure for determining what is wrong, it should be easy for a developmental psychologist to pick and choose from the many strategies that have been developed to help a person become much more balanced.
What Are Some Strategies That Work?
One of the initial strategies that is implemented is focusing on the three goals. These goals include allowing the person to describe what they believe is going on, explain why they think it is going on, and then take that information and optimize a developmental plan of action. The reason that they need to describe what is happening is that it looks at normative development or the typical pattern changes that can be seen repeating with people that have similar problems.
The second stage is explaining why they think something is happening. The psychologist can then use what is called nature and nurture. Nature is about biological processes and nurture refers to a person's environment and their emotions. Finally, the psychologist will be able to look at everything that has been stated and come up with the treatment plan on how to make definitive changes to help the people that are currently unable to resolve certain issues.
Some people trace this all the way back to Charles Darwin when he was talking about scientific observations of Doddy his infant son. Later on, a German psychologist by the name of Wilhelm Preyer started focusing on the mind of children, specifically looking at the developmental processes of his daughter. Empirical studies were done and by the 1990s, the theory of human development was formed. Notables such as Jean Piaget and Vygotsky were able to create many of the foundational theories that are used in developmental psychology.
The main reason that this is able to help people, sometimes immediately, is that the individuals that sit down with a developmental psychologist are able to talk everything out. Once they are able to verbalize this, and if they believe that the other person is able to help them, they are going to be open and honest about how they feel right now. This is the information that a developmental psychologist will need in order to implement the proper strategies to resolve their issues. It's all about trust, and proven psychological techniques that allow people to sometimes see prompt results once they understand the developmental processes and strategies to help.
Developmental psychology is something that is helpful for almost everyone. If a person is able to understand why they are thinking or reacting in a certain way, it can help them understand what is causing the problem. This brief discussion about what is developmental psychology should show you that this is one of the effective ways to start understanding how to resolve problems that people may have with each other. It may also help people and problems that they are facing on their own once they know why they are thinking or feeling the way that they do.


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Philosophy East and West, vol. 54, no. 1 (2004)
Posted on January 23, 2004 by UH Press
Kōans in the Dōgen Tradition: How and Why Dōgen Does What He Does with Kōans
Steven Heine, 1
A hallmark of Dōgen's legacy is his introduction of Chinese Ch'an kōan literature to Japan in the first half of the thirteenth century and his unique and innovative style of interpreting dozens of kōan cases, many of which are relatively obscure or otherwise untreated in the annals. What constitutes the distinctiveness of Dōgen's approach? According to Hee-Jin Kim's seminal study, Dōgen shifts from an instrumental to a realizational model of kōan interpretation. While this essay agrees with some features of Kim's approach, especially his emphasis on the importance of language, it is argued that Kim overlooks the diversity of aims and intentions in Dōgen's use of rhetorical and narrative strategies to highlight diverse doctrinal and ritual themes. There is no single underlying view of kōans for Dōgen, who continually modifies his interpretive approach to particular cases in order to articulate specific themes.
Dharmakīrti and Priest on Change
Chris Mortensen, 20
Competing accounts of change and motion are given by the seventh-century Buddhist logician Dharmakīrti and the contemporary analytical philosopher Graham Priest. They agree on much, but disagree on the issue of the Law of Non-Contradiction. Priest takes Dharmakīrti's side, appealing to current space-time theory, while making some qualifications.
'Place' and 'Being-time': Spatiotemporal Concepts in the Thought of Nishida Kitarō and Dōgen Kigen
Rein Raud, 29
Presented here is a comparative analysis of spatiotemporal concepts in the thought of Nishida and Dōgen, arguing that both thinkers articulate fundamental notions about being and self/subject through them. It starts with an analysis of the notions of 'world' (sekai) and 'place' (basho) as well as time and order in Nishida's work, which is followed by an effort to elucidate his philosophical position by comparing his views to those of Dōgen and by demonstrating their similarity in several important aspects, thus showing that Nishida's philosophy, regardless of its debt to Western influence, is firmly embedded in the tradition of Asian thought.
From Cannibalism to Empowerment: An Analects-inspired Attempt to Balance Community and Liberty
Sor-hoon Tan, 52
Developed here is a Confucian balance between two key democratic ideals, liberty and community, by focusing on the Confucian notion of li (ritual), which has often been considered hostile to liberty. By adopting a semiotic approach to li and relating it to recent studies of ritual in various Western disciplines, li's contribution to communication and its aesthetic dimension are explored to show how emphasizing harmony without sacrificing reflective experience and personal fulfillment renders li a concept of moral empowerment of free individuals in community.
Ibn Sīnā and Husserl on Intention and Intentionality
Marina Paola Banchetti-Robino, 71
The concepts of intention and intentionality were particularly significant notions within the Christian, Jewish, and Islamic medieval philosophical traditions, and they regained philosophical importance in the twentieth century. The theories of intention and intentionality of the medieval Islamic philosopher and physician Ibn Sīnā and the phenomenological philosopher and mathematician Edmund Husserl are examined, compared, and contrasted here, showing that Ibn Sīnā's conception of intention is naturalistic and, in its naturalism, is influenced by the medical professional culture to which Ibn Sīnā belonged. As well, Husserl's anti-naturalistic conception of intentionality is influenced by his background as a mathematician and by his desire to ground mathematics and the empirical sciences in a truly scientific philosophy. In conclusion, an argument is presented for the superiority of the Husserlian transcendentalist account of intentionality over the Avicennian naturalistic account, on the grounds that the latter falls prey to psychologism and reductionism, the two specters that according to Husserl must haunt all naturalistic accounts of consciousness.
FEATURE REVIEW
The Shifting Contours of the Confucian Tradition, a review of Imagining Boundaries: Changing Confucian Doctrines, Texts, and Hermeneutics, edited by Kai-wing Chow, On-cho Ng, and John Henderson
Philip J. Ivanhoe, 83
The Concept of Bodhicitta in Śāntideva's Bodhicaryāvatāra, by Francis Brassard
Reviewed by William Edelglass, 95
Evil and/or/as The Good: Omnicentrism, Intersubjectivity, and Value Paradox in Tiantai Buddhist Thought, by Brook Ziporyn
Reviewed by David R. Loy, 99
Varieties of Religion Today: William James Revisited, by Charles Taylor
Reviewed by Ralph Weber, 103
Persons and Valuable Worlds: A Global Philosophy, by Eliot Deutsch
Reviewed by Joel J. Kupperman, 106
Posted in Philosophy East and West
Pacific Science, vol. 58, no. 1 (2004)
Asian Theatre Journal, vol. 21, no. 1 (2004)


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SNK Solicitors are based at Atlantic Business Centre a few minutes from Altrincham Town Centre and the main transport interchange.
The centre is close to the A56, linking up with the M60 to the north and the M56 to the south.
The closest stations are Navigation Road, which is a 16-minute walk away via Navigation Road and Manchester Road, or Altrincham station, which is a 13-minute bus ride away.
Navigation Road - 0.7 miles
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The business centre is easily accessible by bus 263
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Supported by NIA AG016573. J.D.G. and D.L.G. were supported by NIA 1R21AG056931. J.D.G. was supported by UL1 TR000153.
Reprints: Joshua D. Grill, PhD, 3204 Biological Sciences III, University of California Irvine, Irvine, CA 92697-4545 (e-mail: jgrill@uci.edu).
Alzheimer Disease & Associated Disorders32(4):366-368, October-December 2018.