Convergence of a continuous Galerkin method for hyperbolic-parabolic systems

TL;DR

This paper analyzes a continuous Galerkin finite element method for coupled hyperbolic-parabolic systems, providing optimal error estimates and confirming them through numerical experiments, with implications for advanced multi-physics modeling.

Contribution

It develops and proves optimal error estimates for a space-time Galerkin method applied to coupled hyperbolic-parabolic systems, including complex coupling control and higher order discretizations.

Findings

Optimal order error estimates are established.

Numerical experiments confirm theoretical results.

Higher order discretizations lead to complex algebraic systems.

Abstract

We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modeling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in time and inf-sup stable pairs of finite element spaces for the spatial variables are investigated. Optimal order error estimates are proved by an analysis in weighted norms that depict the energy of the system's unknowns. A further important ingredient and challenge of the analysis is the control of the couplings terms. The techniques developed here can be generalized to other families of Galerkin space discretizations and advanced models. The error estimates are confirmed by numerical experiments, also for higher order piecewise polynomials in time and space. The latter lead to algebraic systems with complex block structure and put a facet of challenge on the design of iterative solvers. An efficient solution technique is referenced.