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Structure preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system

Markus Bause, Sebastian Franz

TL;DR

This work presents a space-time discontinuous Galerkin framework for a prototypical coupled hyperbolic–parabolic system arising in poroelasticity and thermoelasticity. The authors recast the model as a first-order evolutionary problem $(\partial_t M_0 + M_1 + A)U = F$ in the weighted Bochner space $H_\nu(\mathbb{R};H)$ and apply Picard's theorem to obtain well-posedness. A structure-preserving DG discretization in space and time is developed: DG gradients/divergences on broken spaces are defined with boundary corrections to restore skew-selfadjointness of the discrete operator $A_h$, and a fully discrete nonconforming scheme with penalty and boundary terms is analyzed. The paper proves well-posedness of the fully discrete scheme and derives error estimates in the mesh-dependent $\|\cdot\|_{\tau,\nu}$ and the $H_\nu(\mathbb{R};H)$ norms, showing convergence with rates dependent on the time polynomial degree $k$ and the spatial degree $r$. These results provide a robust, structure-preserving method for accurate flux and stress approximations in coupled hyperbolic–parabolic models, with potential applications in poroelasticity and thermoelasticity simulations.

Abstract

We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by the unified abstract solution theory of R. Picard. 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 first-order differential operator in space is recovered. Well-posedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.

Structure preserving discontinuous Galerkin approximation of a hyperbolic-parabolic system

TL;DR

This work presents a space-time discontinuous Galerkin framework for a prototypical coupled hyperbolic–parabolic system arising in poroelasticity and thermoelasticity. The authors recast the model as a first-order evolutionary problem in the weighted Bochner space and apply Picard's theorem to obtain well-posedness. A structure-preserving DG discretization in space and time is developed: DG gradients/divergences on broken spaces are defined with boundary corrections to restore skew-selfadjointness of the discrete operator , and a fully discrete nonconforming scheme with penalty and boundary terms is analyzed. The paper proves well-posedness of the fully discrete scheme and derives error estimates in the mesh-dependent and the norms, showing convergence with rates dependent on the time polynomial degree and the spatial degree . These results provide a robust, structure-preserving method for accurate flux and stress approximations in coupled hyperbolic–parabolic models, with potential applications in poroelasticity and thermoelasticity simulations.

Abstract

We study the numerical approximation of a coupled hyperbolic-parabolic system by a family of discontinuous Galerkin space-time finite element methods. The model is rewritten as a first-order evolutionary problem that is treated by the unified abstract solution theory of R. Picard. 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 first-order differential operator in space is recovered. Well-posedness of the fully discrete problem and error estimates for the discontinuous Galerkin approximation in space and time are proved.
Paper Structure (22 sections, 11 theorems, 135 equations)

This paper contains 22 sections, 11 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Evolutionary formulation and its well-posedness
  3. Proof.
  4. Discontinuous Galerkin discretization and well-posedness
  5. Notation and auxiliaries
  6. Gradient and divergence on broken function spaces
  7. Gradient and divergence on broken polynomial spaces and operator $\boldsymbol{A_h}$
  8. Proof.
  9. Fully discrete problem of structure preserving nonconforming approximation
  10. Proof.
  11. Error estimation for structure preserving nonconforming approximation
  12. Proof.
  13. Proof.
  14. Proof.
  15. Proof.
  16. ...and 7 more sections

Key Result

Theorem 2.5

Let $H$ denote a real Hilbert space. Let $M_0,M_1:H\rightarrow H$ be bounded linear selfadjoint operators and $A:D(A)\subset H\rightarrow H$ skew-selfadjoint. Moreover, suppose that there exists some $\nu_0 >0$ such that Then, for each $\nu \geq \nu_0$ and each $F\in H_\nu(\mathbb{R};H)$ there exist a unique solution $U\in H_{\nu}(\mathbb{R};H)$ such that where the closure is taken in $H_\nu(\ma

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.5: Well-posedness
  • Corollary 2.6: Well-posedness of Problem \ref{['Prob:EP']}
  • Remark 2.7
  • Definition 3.1: DG derivatives
  • Lemma 3.2
  • Definition 3.3: DG derivatives for single-valued functions
  • Lemma 3.4: DG skew-selfadjointness
  • ...and 10 more