Refinement of flexible space–time finite element meshes and discontinuous Galerkin methods

In this paper we present an algorithm to refine space–time finite element meshes as needed for the numerical solution of parabolic initial boundary value problems. The approach is based on a decomposition of the space–time cylinder into finite elements, which also allows a rather general and flexible discretization in time. This also includes adaptive finite element meshes which move in time. For the handling of three-dimensional spatial domains, and therefore of a four-dimensional space–time cylinder, we describe a refinement strategy to decompose pentatopes into smaller ones. For the discretization of the initial boundary value problem we use an interior penalty Galerkin approach in space, and an upwind technique in time. A numerical example for the transient heat equation confirms the order of convergence as expected from the theory. First numerical results for the transient Navier–Stokes equations and for an adaptive mesh moving in time underline the applicability and flexibility of the presented approach.

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