Convergence of a continuous Galerkin method for hyperbolic-parabolic systems

We study the numerical approximation by space-time finite element methods of a coupled hyperbolic -parabolic system modeling, for instance, 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 for the first -order energy of the system's variables. An important ingredient of the estimates is the control of the coupling terms by a tailored testing strategy. The techniques developed here can be generalized to other families of space-time finite element discretizations and related models. The error estimates are confirmed by numerical experiments, also for higher order piecewise polynomials in space and time. 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.