Space-time hexahedral finite element methods for parabolic evolution problems

TL;DR

This paper introduces locally stabilized, conforming space-time finite element methods on hexahedral meshes for parabolic problems, with anisotropic error estimates and adaptive refinement strategies demonstrated through numerical experiments.

Contribution

It develops a new space-time finite element framework with anisotropic error estimates and adaptive strategies for parabolic equations on hexahedral meshes.

Findings

Explicit anisotropic error estimates derived

Effective anisotropic adaptive mesh refinement strategies implemented

Numerical experiments validate the proposed methods

Abstract

We present locally stabilized, conforming space-time finite element methods for parabolic evolution equations on hexahedral decompositions of the space-time cylinder. Tensor-product decompositions allow for anisotropic a priori error estimates, that are explicit in spatial and temporal meshsizes. Moreover, tensor-product finite elements are suitable for anisotropic adaptive mesh refinement strategies provided that an appropriate a posteriori discretization error estimator is available. We present such anisotropic adaptive strategies together with numerical experiments.