Volume 63, pp. 1-32, 2025.

Structure-preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system

Markus Bause, Sebastian Franz, and Mathias Anselmann

Abstract

We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin (DG) space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by a unified abstract solution theory. For the discretization in space, generalizations of the distribution gradient and divergence operators on broken polynomial spaces are defined. Since their skew-selfadjointness is perturbed by boundary surface integrals, adjustments are introduced such that the skew-selfadjointness of the discrete counterpart of the total system's first-order differential operator in space is recovered. Well-posedness of the fully discrete problem and error estimates for the DG approximation in space and time are proved.

Full Text (PDF) [450 KB], BibTeX

Key words

Coupled hyperbolic-parabolic problem, first-order system, Picard's theorem, discontinuous Galerkin space-time discretization, error estimates

AMS subject classifications

65M15, 65M60, 35Q35

Links to the cited ETNA articles

[32]	Vol. 54 (2021), pp. 198-209 Sebastian Franz: Continuous time integration for changing type systems

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