Biharmonic equation
As an example for the treatment of higher order elliptic equations we consider the two dimensional biharmonic equation \begin{equation} \left\{\begin{aligned} \triangle^2u=f &~~~~~~ \mathrm{in}~ \Omega\subset\mathbb{R}\\ u=\frac{\partial u}{\partial n}=0 &~~~~~~ \mathrm{on}~ \Gamma. \end{aligned}\right.\label{eqn:373} \end{equation} which models the vertical displacement of the mid-surface of a clamped plate. Problem (\ref{eqn:373}) fits into the abstract framework of Section 2.1 with $$X=Y:=\{u\in W^{2,2}(\Omega): u=\frac{\partial u}{\partial \mathbf{n}}=0~~\mathrm{on}~ \Gamma\},$$ $$\|\cdot\|_X=\|\cdot\|_Y=\|\cdot\|_{2,2},$$ \begin{equation} \langle F(u),\varphi\rangle:=\int_{\Omega}\triangle u\triangle\varphi-\int_{\Omega}f\varphi.\label{eqn:374} \end{equation} Since the bilinear form ~$u,v\rightarrow\int_{\Omega}\triangle u\triangle v$~ is continuous and coercive on ~$X$(cf.[37]), we have ~$DF(u)\in \mathrm{Isom}(X,Y^*)$~ for all $u\in X$.
For the discretization of Problem (\ref{eqn:373}) we assume that $X_h\subset X$ and $Y_h\subset Y$ are finite element spaces corresponding to $\mathcal{T}_h$ and consisting of piecewise polynomials.These conditions in particular imply that the functions in $X_h$ and ~$Y_h$~ are of class ~$C^1$.~ Examples of ~$C^1$-conforming finite element spaces are given in, e.g., [37;Chapter 6]. Denote by $k\geq$ the maximal polynomial degree of the functions in $X_h$. We denote by ~$J_h\rightarrow Y_h$ the Cl$\mathrm{\'{e}}$ment interpolation operator which satisfies the following analogues of Lemma 1.4 and 3.1 [cf. 38] \begin{equation} \|\varphi-J_h\varphi\|_{0,2;T}\leq c_1h_T^2\|\varphi\|_{2,2;\tilde{\omega}_T}\quad \forall \varphi\in W^{2,2}(\tilde{\omega}_T), T\in \mathcal{T}_h \label{eqn:375} \end{equation} \begin{equation} \|\varphi-J_h\varphi\|_{2;E}+h_E\|n_E\cdot\nabla(\varphi-J_h\varphi)\|_{2,E} \leq c_2h_E^{3/2}\|\varphi\|_{2,2;\tilde{\omega}_E}\quad \forall \varphi\in W^{2,2}(\tilde{\omega}_E), E\in \mathcal{E}_{h,\Omega} \label{eqn:376} \end{equation}
We set $R_h:=J_h$ and define $\tilde{F}_h$ in the same way as $F$ with $f$ replaced by $$f_h:=\sum_{T\in\mathcal{T}_h}\pi_{l,T}f.$$ Here, $l\geq 0$ is an arbitrary integer which will be kept fixed in what follows.
Using integration by parts elementwise, we obtain for all $\varphi\in Y$ and $u_h\in X_h$ \begin{equation} \langle \tilde{F}(u_h),\varphi\rangle=\sum_{T\in\mathcal{T}_h}\int_T(\triangle^2u_h-f)\varphi+\sum_{E\in\mathcal{E}_{h,\Omega}}\{\int_E[\triangle u_h]_En_E\cdot\nabla\varphi-\int_E[n_E\cdot\nabla\triangle u_h]_E\varphi\} \label{eqn:377} \end{equation} and \begin{equation} \langle \tilde{F}_h(u_h),\varphi\rangle=\sum_{T\in\mathcal{T}_h}\int_T(\triangle^2u_h-f_h)\varphi+\sum_{E\in\mathcal{E}_{h,\Omega}}\{\int_E[\triangle u_h]_En_E\cdot\nabla\varphi-\int_E[n_E\cdot\nabla\triangle u_h]_E\varphi\} \label{eqn:378} \end{equation}
Equation (\ref{eqn:377}) and (\ref{eqn:378}) together with Inequality (\ref{eqn:375}) immediately imply
\begin{equation} \begin{aligned} &\|(Id_Y-R_h)^*[\tilde{F}_h(u_h)]\|_{Y^*}\\ =&\sup_{\varphi\in Y,\|\varphi\|_Y=1}\sum_{T\in\mathcal{T}_h}\int_T(f-f_h)(\phi-J_h\varphi)\\ \leq&c_1\{\sum_{T\in\mathcal{T}_h}h^4_T\|f-f_h\|^2_{0,2,T}\}^{1/2}\\ =&c_1\{\sum_{T\in\mathcal{T}_h}\varepsilon^2_T\}^{1/2}, \end{aligned}\label{eqn:379} \end{equation} where we have set for abbreviation \begin{equation} \varepsilon_T:=h^2_T\|f-f_h\|_{0,2;T}\quad \forall T\in \mathcal{T}_h.\label{eqn:380} \end{equation} Equations (\ref{eqn:378}) and Estimates (\ref{eqn:375}),(\ref{eqn:376}) on the other hand yield \begin{equation} \begin{aligned} &\|(Id_Y-R_h)*\tilde{F}_h(u_h)\|_{Y^*}\\ =&\sup_{\phi\in Y,\|\varphi\|_Y=1}\{\sum_{T\in\mathcal{T}_h}\int_T(\triangle^2u_h-f_h)(\varphi-J_h\varphi)\\ &\quad + \sum_{E\in\mathcal{E}_{h,\Omega}}\{\int_E[\triangle u_h]_En_E\cdot\nabla(\varphi-J_h\varphi)-\int_E[n_E\cdot\nabla\triangle u_h]_E(\phi-J_h\varphi)\}\}\\ \leq&c\{\sum_{T\in\mathcal{T}_h}h_T^4\|\triangle^2u_h-f_h\|^2_{0,2;T}\\ &\quad + \sum_{E\in\mathcal{E}_{h,\Omega}}[h_E\|[\triangle u_h]_E\|^2_{2;E}+h_E^3\|[n_E\cdot\nabla\triangle u_h]_E\|_{2;E}\}^{1/2}\\ \leq&c'\{\sum_{T\in\mathcal{T}_h}\eta_T^2\}^{1/2},\label{eqn:381} \end{aligned} \end{equation} where we have set for abbreviation \begin{equation} \begin{aligned} \eta_T:= &\{h_T^4\|\triangle^2u_h-f_h\|^2_{0,2;T}\\ &\quad +\sum_{E\in\mathcal{E}(T)\subset \mathcal{E}_{h,\Omega}} [h_E]\|[\triangle u_h]_E\|^2_{2;E} + h_E^3\|[n_E\cdot\nabla\triangle u_h]_E\|^2_{2;E}]\}^{1/2}\\ & \forall T\in \mathcal{T}_h. \end{aligned}\label{eqn:382} \end{equation} Relations (\ref{eqn:378}),(\ref{eqn:381}),and (\ref{eqn:382}) and the results of Setions 3.1 and 3.3 show that the space $\tilde{Y}_h$ must balance the following quantities:
on each triangle:~~$\triangle^2u-f_h$,
on each interior edge: ~~ $[\triangle u_h]_E$ and $[n_E\cdot \nabla\triangle u_h]_E$.
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To achieve this, we adopt the strategy of Section 3.1 and 3.3 and make the following Ansatz \begin{equation} \tilde{Y}_h:=\mathrm{span}\{\varphi_Tv,~\varphi_{E,1}P_1\sigma,~\varphi_{E,2}P_2\tau:~v\in\Pi_{m_0\mid_{T}},~\sigma\in\Pi_{m_1\mid_E},~T\in\mathcal{T}_h,~E\in\mathcal{E}_{h,\Omega}\} \label{eqn:383} \end{equation} where $m_0:=\max\{k-4,l\},~m_1:=\max\{k-2,0\}$ and $m_2:=\max\{k-3,0\}$.
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The functions $\varphi_T,\varphi_{E,1}$, and $\varphi_{E,2}$ and the prolongation operators $P_1$ and $P_2$ will be constructed in what follows. In doing this we have to keep in mind that, due to the condition $\tilde{Y} \subset Y\subset W^{2,2}(\Omega)$, the functions in $\tilde{Y}_h$ must be continuously differentiable.
Consider first an arbitary $T\in\mathcal{T}_h$ and define $$\varphi_T:=b_T^2,$$ where $b_T$ is the triangle-bubble function introduced in Section 1.1. Since $\varphi_T$ together with its first order derivatives vanishes on $\partial T$, we have $\varphi_Tv\in Y$ for all $v\in \Pi_{m_0|T}$. The arguments used in the proof of Lemma 3.3 imply for all $v\in \Pi_{m_0|T}$ $$c_1\|v\|^2_{0,2,T}\leq\int_T\varphi_Tv^2,$$ $$\|\varphi_Tv\|_{0,2;T}\leq \|v\|_{0,2;T},$$ $$c_2h_T^{-2}\|\varphi_Tv\|_{0,2;T}\leq\|\varphi_Tv\|_{2,2,T}\leq c_3h_T^{-2}\|\varphi_Tv\|_{0,2;T},$$ where the constants $c_1,c_2,c_3$ depend on $m_0$ and $h_T/\varrho$. Put $$w_T:=\phi_T(\triangle^2u_h-f_h).$$ The above estimates and Equation (\ref{eqn:378}) then imply \begin{equation*} \begin{aligned} c_1\|\|_{0,2;T}^2&\leq\int_T(\triangle^2u_h-f_h)w_T\\ &=\langle\tilde{F}_h(u_h),w_T\rangle\\ &\leq c_3 h_T^{-2}\|\triangle^2u_h-f_h\|_{0,2;T}\sup_{\varphi\in\tilde{Y}_{h|T},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle \end{aligned} \end{equation*} and thus \begin{equation} h_T^2\|\triangle^2u_h-f_h\|_{0,2;T}\leq c\sup_{\varphi\in\tilde{Y}_{h|T},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle.\label{eqn:384} \end{equation}
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Next we consider an arbitrary $E\in\mathcal{E}_{h,\Omega}$. Let $T_1,T_2\in\mathcal{T}_h$ be such that $\omega_E=T_1\cup T_2$ and that $n_E$ is the exterior normal to $T_1$. The orientation of $E$ is chosen such that $T_1$ is lying on the left when passing through $E$ in this orientation. Set $$\varphi_{E,1}:=(b_{T_1}-b_{T_2})b_E~~\text{and}~~P_1:=P,$$ where $b_E$ is the edge-bubble function introduced in Section 1.1 and where $P$ is the prolongation operator of Section 3.1. Consider an arbitrary $\sigma\in\Pi_{m_1|E}$. An easy calculation shows that $\varphi_{E,1}P_{\sigma}$ is continuously differentiable across $E$ and that \begin{equation} n_E\cdot\nabla(\varphi_{E,1}P\sigma)=\frac{27}{8}\big(\frac{h_E}{|T_1|}+\frac{h_E}{|T_2|}\big)b_E^2P\sigma\label{eqn:385} \end{equation} holds on E. Since $\varphi_{E,1}P\sigma$ together with its first order devivatives vanishes on $\partial\omega_E$, we conclude that $\varphi_{E,1}P\sigma\in Y$. Moreover, Equation (\ref{eqn:385}) and the arguments used in the proof of Lemma 3.3 imply \begin{equation*} \begin{aligned} c_4h_E^{-1}\|\sigma\|^2_{2;E}\leq&\int_E\sigma n_E\cdot\nabla(\varphi_{E,1}P\sigma),\\ \|n_E\cdot\nabla(\varphi_{E,1}P\sigma)\|_{2;E}\leq& c_5h_E^{-1}\|\sigma\|_{2;E},\\ c_6h_E^{-2}\|\varphi_{E,1}P\sigma\|_{0,2;\omega_E}\leq&\|\varphi_{E,1}P\sigma\|_{2,2;\omega_E}\leq c_7 h_E^{-2}\|\varphi_{E,1}P\sigma\|_{0,2;\omega_E},\\ \|\varphi_{E,1}P\sigma\|_{0,2;\omega_E}\leq& c_8h_E^{1/2}\|\sigma\|_{2;E}, \end{aligned} \end{equation*} where the constants $c_4,\cdots,c_8$ depend on $m_1$ and $h_T/\varrho_T$. Put$$w_{E,1}:=\varphi_{E,1}P([\triangle u_h]_E).$$ The above estimates, Equations (\ref{eqn:378}),(\ref{eqn:375}),and Inequality (\ref{eqn:378}) then yield \begin{equation*} \begin{aligned} c_4h_E^{-1}\|[\triangle u_h]_E\|_{2;E}^2 &\leq\int_E[\triangle u_h]_En_E\cdot\nabla w_{E,1}\\ & =\langle\tilde{F}_h(u_h),w_{E,1}\rangle-\sum{i=1,2}\int_{T_i}(\triangle^2u_h-f_h)w_{E,1}\\ &\leq h_E^{-2}\|w_{E,1}\|_{0,2;\omega_E}\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle\\ &\leq c_8h_E^{-3/2}\|[\triangle u_h]_E\|_{2;E}\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle \end{aligned} \end{equation*} and thus \begin{equation} h_E^{1/2}\|[\triangle u_h]_E\|_{2;E}\leq c'\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle.\label{eqn:386} \end{equation} In order to balance the $[n_E\cdot\nabla\triangle u_h]_E$-term, we use Remark 3.6 and set $\varphi_{E,2}:=\psi_E,P_2:=\tilde{P}$, and $$w_{E,2}:=\psi_E\tilde{P}([n_E\cdot\nabla\triangle u_h]_E).$$ Remark 3.6, Equation (\ref{eqn:378}), and Inqualities (\ref{eqn:374}) and (\ref{eqn:376}) then give \begin{equation*} \begin{aligned} &c_9\|[n_E\cdot\nabla\triangle u_h]_E\|_{2;E}^2\\ \leq&\int_E[n_E\cdot\nabla\triangle u_h]_Ew_{E,2}\\ =&\sum_{i=1,2}\int_{T_i}(\triangle^2u_h-f_h)w_{E,2}+\int_E[\triangle u_h]_En_E\cdot\nabla w_{E,2}-\langle\tilde{F}_h(u_h),w_{E,2}\rangle\\ \leq&\|[n_E\cdot\nabla\triangle u_h]_E\|_{2;E}\bigg\{c_{13}\bigg[\sum_{i=1,2}h_E\|\triangle^2u_h-f_h\|_{0,2;T_i}^2\bigg]^{1/2}\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~+c_{12}h_E^{-1}\|[\triangle u_h]_E\|_{2;E}\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~+c_{11}c_{13}h_E^{-3/2}\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle\bigg\}\\ \leq&ch_E^{-3/2}\|[n_E\cdot\nabla\triangle u_h]_E\|_{2;E}\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle \end{aligned} \end{equation*} and thus \begin{equation} h_E^{3/2}\|\|_{2;E}\leq c'\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle. \label{eqn:387} \end{equation} Inequalities (\ref{eqn:384}), (\ref{eqn:386}), and (\ref{eqn:387}) imply for all $T\in\mathcal{T}_h$ \begin{equation} \eta_T\leq c\sup_{\varphi\in\tilde{Y}_{h|w_E},\|\varphi\|_Y=1}\langle\tilde{F}_h(u_h),\varphi\rangle.\label{eqn:388} \end{equation} Using the same arguments as in the preceding sections this together with Inequatity (\ref{eqn:381}) prove Estimate (2.11). Finally, the above estimates for the functions in $\tilde{Y}_h$ and Equations (\ref{eqn:377}) and (\ref{eqn:378}) imply in the usual way that \begin{equation} \|\tilde{F}_h(u_h)-F(u_h)\|_{\tilde{Y}_h^*}\leq\bigg\{\sum_{T\in\mathcal{T}_h}\varepsilon_T^2\bigg\}^{1/2}\label{eqn:389} \end{equation} and \begin{equation} \|\tilde{F}_h(u_h)\|_{\tilde{Y}_h^*}\leq\bigg\{\sum_{T\in\mathcal{T}_h}\eta_T^2\bigg\}^{1/2}\label{eqn:390} \end{equation}
Proposition 2.1 and 2.5 and Inequalities (\ref{eqn:379})--(\ref{eqn:390}) therefore yield the following a posteriori error estimate for Problem (\ref{eqn:373}). {\textbf Proposition 3.30.} Let $u\in X$ be the unique weak solution of Problem (\ref{eqn:373}),i.e. of F(u)=0, and let $u_h\in X_h$ be an approximate solution of the corresponding discrete problem. Then the following a posteriori error estimates hold \begin{equation*} \begin{aligned} \|u-u_h\|_{2,2}\leq&c_1\bigg\{\sum_{T\in\mathcal{T}_h}\eta_T^2\bigg\}^{1/2}+c_2\bigg\{\sum_{T\in\mathcal{T}_h}\varepsilon_T^2\bigg\}^{1/2}\\ &+c_3\|F(u_h)-F_h(u_h)\|_{Y^*}+c_4\|F_h(u_h)\|_{Y^*} \end{aligned} \end{equation*} and$$\eta_T\leq c_5\|u-u_h\|_{2,2;\omega_T}+c_6\bigg\{\sum_{T'\in\omega_T}\varepsilon_{T'}^2\bigg\}^{1/2}\quad \forall T\in\mathcal{T}_h.$$ Here, $\varepsilon_T$ and $\eta_T$ are given by Equations (\ref{eqn:380}) and (\ref{eqn:382}), $\|F(u_h)-F_h(u_h)\|_{Y_h^*}$ and $\|F_h(u_h)\|_{Y_h^*}$ denote the consistency error of the dicretization and the residual of the discrete problem, and the constants $c_1,\cdots,c_6$ depend on $h_T/\varrho_T$ and the integers $k$ and $l$.
Remark 3.31. In order to avoid the use of $C^1$-elements, one often considers mixed finite element approximations of Problem (\ref{eqn:373}) which are obtained by discretizing the weak formulation of an appropriate equivalent elliptic system of 2nd order. Since these mixed finite element discretizations only involve $C^0$-elements, these can be analyzed along the lines of Sections 3.1,3.3,and 3.5.
Remark 3.32. Equations (\ref{eqn:377}) and (\ref{eqn:378}) rely on the fact that $u_{h|T}\in W^{4,2}(T)$ for all $T\in\mathcal{T}_h$. When using composite elements, one has to perform the integration by parts on each composite seperately. This gives rise to additional jump terms across the boundaries of the composites which may be handled in the same way as jump terms in Equation (\ref{eqn:377}) and (\ref{eqn:378}).
Remark 3.33. To our knowledge, a posteriori error estimators for $4$th order problems have not been considered before.
