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Time-dependent Dirichlet conditions in finite element discretizations

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      Abstract

      For the modeling and numerical approximation of problems with time-dependent Dirichlet boundary conditions, one can call on several consistent and inconsistent approaches. We show that spatially discretized boundary control problems can be brought into a standard state space form accessible for standard optimization and model reduction techniques. We discuss several methods that base on standard finite element discretizations, propose a newly developed problem formulation, and investigate their performance in numerical examples. We illustrate that penalty schemes require a wise choice of the penalization parameters in particular for iterative solves of the algebraic equations. Incidentally, we confirm that standard finite element discretizations of higher order may not achieve the optimal order of convergence in the treatment of boundary forcing problems and that convergence estimates by the common method of manufactured solutions can be misleading.

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      Most cited references 32

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      Optimal Control of Systems Governed by Partial Differential Equations

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        Boundary Control of PDEs

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          Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

           J.-P. Raymond (2007)
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            Author and article information

            Affiliations
            [1 ]Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany
            Author notes
            [* ]Corresponding author’s e-mail address: heiland@ 123456mpi-magdeburg.mpg.de
            Contributors
            Journal
            SOR-MATH
            ScienceOpen Research
            ScienceOpen
            2199-1006
            06 October 2015
            : 0 (ID: bb5f1992-da64-45fc-bbae-141ff8be018c )
            : 0
            : 1-18
            3126:XE
            10.14293/S2199-1006.1.SOR-MATH.AV2JW3.v1
            © 2015 P. Benner and J. Heiland

            This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

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            Figures: 7, Tables: 6, References: 35, Pages: 18
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