Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

Authors:
Sven Beuchler;Clemens Pechstein;Daniel Wachsmuth
Affiliations:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria 4040;Institute of Computational Mathematics, Johannes Kepler University, Linz, Austria 4040;Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria 4040
Venue:
Computational Optimization and Applications
Year:
2012

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Abstract

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error $\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)}$ decreases like N 驴1, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N 驴3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results.