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

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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 −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 −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.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, 4040, Linz, Austria

    Sven Beuchler & Daniel Wachsmuth

  2. Institute of Computational Mathematics, Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria

    Clemens Pechstein

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  1. Sven Beuchler
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  2. Clemens Pechstein
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  3. Daniel Wachsmuth
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Correspondence to Daniel Wachsmuth.

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Beuchler, S., Pechstein, C. & Wachsmuth, D. Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs. Comput Optim Appl 51, 883–908 (2012). https://doi.org/10.1007/s10589-010-9370-2

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