A Space–Time Finite Element Method for the Linear Bidomain Equations

Steinbach, Olaf; Yang, Huidong

doi:10.1007/978-3-030-14244-5_16

Cited by 2 publications

(1 citation statement)

References 22 publications

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“…This can be seen as an extension of the Petrov-Galerkin space-time finite element method proposed in [32] for parabolic problems, and in our recent work [25] for parabolic optimal control problems. This kind of unstructured space-time finite element approaches has gained increasing interest; see, e.g., [2,3,22,24,34,36,39], and the survey article [35].…”

Section: Introductionmentioning

confidence: 99%

Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations

Langer¹,

Steinbach²,

Tröltzsch³

et al. 2020

Preprint

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We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic trackingtype functional including a Tikhonov regularization term and of the L 1norm of the control that accounts for its spatio-temporal sparsity. We use a space-time Petrov-Galerkin finite element discretization for the firstorder necessary optimality system of the associated discrete optimal sparse control problem. The discretization is based on a variational formulation that employs piecewise linear finite elements simultaneously in space and time. Finally, the discrete nonlinear optimality system that consists of coupled forward-backward state and adjoint state equations is solved by a semismooth Newton method.

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Section: Introductionmentioning

confidence: 99%

Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations

Langer¹,

Steinbach²,

Tröltzsch³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Unstructured space-time finite element methods for optimal control of parabolic equations

Langer

Steinbach

Tröltzsch

et al. 2020

Preprint

View full text Add to dashboard Cite

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babuška's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control constraints. This is done for both continuous and discrete levels. Based on these results, we derive discretization error estimates. Then we consider a semilinear parabolic optimal control problem arising from the Schlögl model. The associated nonlinear optimality system is solved by Newton's method, where a linear system, that is similar to the first-order optimality systems considered for the linear model problems, has to be solved at each Newton step. We present various numerical experiments including results for adaptive space-time finite element discretizations based on residualtype error indicators. In the last two examples, we also consider semilinear parabolic optimal control problems with box constraints imposed on the control.

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A Space–Time Finite Element Method for the Linear Bidomain Equations

Cited by 2 publications

References 22 publications

Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations

Unstructured Space-Time Finite Element Methods for Optimal Sparse Control of Parabolic Equations

Unstructured space-time finite element methods for optimal control of parabolic equations

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