Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations

Abstract

This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of $h^2|\ln h|^{3/2}$ is proven for domains with interior angles smaller than $2\pi /3$ using quasi-uniform meshes. For larger interior angles mesh grading techniques are used to get the same order of convergence.

Appendix

Lemma 11

Let Assumption 1 be satisfied. Furthermore, let ${\bar{y}_h}=y_h(\bar{u}_h)$ and $y_h(R_h^{\bar{u}}\bar{u})$ be the solutions of (17) w.r.t $\bar{u}_h$ and $R_h^{\bar{u}}\bar{u}$, respectively. Then there exists a mesh size $h_0>0$ such that for all $h<h_0$ the estimate

$$\begin{aligned} \Vert {\bar{y}_h}-y_h(R_h^{\bar{u}}\bar{u})\Vert _{L^2(\varOmega )}\le c\Vert {\bar{u}_h}-R_h^{\bar{u}}\bar{u}\Vert _{L^1(\varGamma )} \end{aligned}$$

is valid.

Proof

Analogously to the beginning of the proof of Lemma 8, we introduce a dual auxiliary problem and its discrete counterpart by: let $\phi \in H^1(\varOmega )$ be the unique solution of

$$\begin{aligned} a(\phi ,v)+\int \limits _{\varOmega } \alpha \phi v \,dx=\int \limits _{\varOmega } \big ({\bar{y}_h}-y_h(R_h^{\bar{u}}\bar{u})\big )v \,dx\quad \quad \forall v\in H^1(\varOmega ), \end{aligned}$$

with

$$\begin{aligned} \alpha (x)=\left\{ \begin{aligned} \frac{d(x,{\bar{y}_h}(x))-d(x,y_h(R_h^{\bar{u}}\bar{u})(x))}{{\bar{y}_h}(x)-y_h(R_h^{\bar{u}}\bar{u})(x)},&\quad \text {if }{\bar{y}_h}(x)-y_h(R_h^{\bar{u}}\bar{u})(x)\not =0,\\ c_\varOmega ,&\quad \text {otherwise,} \end{aligned} \right. \end{aligned}$$

where $c_\varOmega$ denotes the constant from Assumption 1 (A4). Due to the results of Theorem 7, the approximation properties of $R_h^{\bar{u}}$, the uniform convergence of $\bar{u}_h$ to $\bar{u}$ for all $h<h_0$ with $h_0$ from Theorem 8, and the Lipschitz continuity of the nonlinearity $d$ with respect to the second variable, one can easily check that $\alpha$ is uniformly bounded in $L^\infty (\varOmega )$ independent of the mesh parameter $h$. Furthermore, employing the monotonicity of $d$ according to Assumptions 1 (A3) and (A4) we can conclude $\alpha \ge 0$ in $\varOmega$ and the existence of a subset $E_\varOmega$ of $\varOmega$ with $\alpha >0$ in $E_\varOmega$. Thus, it is classical to show that the problem is well-posed for all $h<h_0$. The corresponding discrete counterpart $\phi _h\in V_h$ is the unique solution of the problem

$$\begin{aligned} a(\phi _h,v_h)\!+\!\int \limits _{\varOmega } \alpha \phi _h v_h \,dx\!=\!\int \limits _{\varOmega } ({\bar{y}_h}-y_h(R_h^{\bar{u}}\bar{u}))v_h \,dx\quad \forall v_h\in V_h. \end{aligned}$$

By means of ${\bar{y}_h}(\bar{u}), y_h(R_h^{\bar{u}}\bar{u})\in V_h$ being solutions of (17) and the definition of $\alpha$, we derive

$$\begin{aligned} \Vert {\bar{y}_h}(\bar{u})\!-\!y_h(R_h^{\bar{u}}\bar{u})\Vert _{L^2(\varOmega )}^2\!&= \!a(\phi _h, {\bar{y}_h}(\bar{u})-y_h(R_h^{\bar{u}}\bar{u}))\!+\!\int \limits _{\varOmega } \alpha \phi _h({\bar{y}_h}(\bar{u})- y_h(R_h^{\bar{u}}\bar{u})) \,dx\\&= a({\bar{y}_h}(\bar{u})-y_h(R_h^{\bar{u}}\bar{u}),\phi _h) +\int \limits _{\varOmega } (d(x,{\bar{y}_h}(\bar{u}))\\&\quad -\,d(x,y_h(R_h^{\bar{u}}\bar{u}))\phi _h \,dx\\&= \int \limits _{\varGamma } ({\bar{u}_h}-R_h^{\bar{u}}\bar{u})\phi _h \,ds. \end{aligned}$$

We continue by the estimates

$$\begin{aligned} \begin{aligned} \int \limits _{\varGamma } ({\bar{u}_h}-R_h^{\bar{u}}\bar{u})\phi _h \,ds&\le \Vert {\bar{u}_h}-R_h^{\bar{u}}\bar{u}\Vert _{L^1(\varGamma )}\Vert \phi _h\Vert _{L^\infty (\varGamma )}\\&\le \Vert {\bar{u}_h}-R_h^{\bar{u}}\bar{u}\Vert _{L^1(\varGamma )}(\Vert \phi _h-\phi \Vert _{L^\infty (\varOmega )} +\Vert \phi \Vert _{L^\infty (\varOmega )})\\&\le c\Vert {\bar{u}_h}-R_h^{\bar{u}}\bar{u}\Vert _{L^1(\varGamma )}\left( h^{1/2-\varepsilon }+1\right) \Vert {\bar{y}_h}-y_h(R_h^{\bar{u}}\bar{u})\Vert _{L^2(\varOmega )}, \end{aligned} \end{aligned}$$

where a standard $L^\infty (\varOmega )$-error estimate (see e.g. (24)) and Lemma 1 together with the embedding $H^{3/2}(\varOmega )\hookrightarrow L^\infty (\varOmega )$ were used. Thus, the assertion is proven. $\square$

Lemma 12

Suppose that the Assumptions (A3) and (A4) are fulfilled. Let $M>0$ and $u\in L^2(\varGamma )$ with $\Vert u\Vert _{L^2(\varGamma )}<M$ be given. Moreover, let $y_h^{v}\in V_h$ be the unique solution of (21) for a given discrete state $y_h(u)$ w.r.t. the right hand side $v$. Then the estimate

$$\begin{aligned} \Vert y_h^{v}\Vert _{L^2(\varOmega )}\le c\Vert v\Vert _{L^1(\varGamma )} \end{aligned}$$

holds true with a constant $c$ which may depend on $M$ but is independent of $u$.

Proof

The proof can be done analogously to the proof of Lemma 11 introducing an appropriate dual problem. $\square$