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Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equation

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Abstract

A mixed finite element discretization of an optimal control problem for the linear wave equation with homogeneous Dirichlet boundary condition is considered. For the temporal discretization, a Petrov–Galerkin scheme is utilized and the Raviart–Thomas finite elements for spatial discretization is used. A priori error analysis is proved for this numerical scheme. A hybridized formulation is proposed and if the Arnold–Brezzi post-processing method is applied, better convergence rates with respect to space are observed. The interchangeability of discretization and optimization holds both for mixed and hybrid formulations. Numerical experiments illustrating the theoretical results are presented using the lowest-order Raviart–Thomas elements.

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Acknowledgements

The authors are grateful to the reviewers for the careful reading and valuable suggestions that led to the improvement of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of the Philippines Baguio, Governor Pack Road, 2600, Baguio, Philippines

    Gilbert Peralta

  2. Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010, Graz, Austria

    Karl Kunisch

  3. RICAM Institute, Austrian Academy of Sciences, Altenbergerstrasse 69, 4040, Linz, Austria

    Karl Kunisch

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  1. Gilbert Peralta
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  2. Karl Kunisch
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Correspondence to Gilbert Peralta.

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Peralta, G., Kunisch, K. Mixed and hybrid Petrov–Galerkin finite element discretization for optimal control of the wave equation. Numer. Math. 150, 591–627 (2022). https://doi.org/10.1007/s00211-021-01258-9

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