Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations

Abstract

We study the numerical approximation of Neumann boundary optimal control problems governed by a class of quasilinear elliptic equations. The coefficients of the main part of the operator depend on the state function, as a consequence the state equation is not monotone. We prove that strict local minima of the control problem can be approximated uniformly by local minima of discrete control problems and we also get an estimate of the rate of this convergence. One of the main issues in this study is the error analysis of the discretization of the state and adjoint state equations. Some difficulties arise due to the lack of uniqueness of solution of the discrete equations. The theoretical results are illustrated by numerical tests.

Appendix

Lemma A.1

There exists a constant C>0 such that

$$\|z\|_{L^{4}(\Omega)}\le C\|z\|_{L^{2}(\Omega)}^{{1}/{2}}\|z\|_{H^{1}(\Omega)}^{{1}/{2}}\quad\forall z\in H^{1}(\Omega).$$

Proof

First we recall the following well known inequality (cf. [29, Lemma 3.3])

$$\|z\|_{L^{4}(\mathbb{R}^2)}\le2^{{1}/{4}}\|z\|_{L^{2}(\mathbb{R}^2)}^{{1}/{2}}\|\nabla z\|_{L^{2}(\mathbb{R}^2)}^{{1}/{2}}\quad\forall z\in H^1\bigl(\mathbb{R}^2\bigr). $$

(A.1)

Now, we use an extention operator E:H ¹(Ω)→H ¹(ℝ²) such that

$$\|Ez\|_{L^{2}(\mathbb{R}^{2})}\le C_{1}\|z\|_{L^{2}(\Omega)}\quad\text{and}\quad\|Ez\|_{H^{1}(\mathbb{R}^{2})}\le C_{2}\|z\|_{H^{1}(\Omega)}; $$

(A.2)

see [26, Theorem 3.9 and Remark 3.5]. Then, (A.1) and (A.2) imply

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