Space-Time CutFEM on Overlapping Meshes: Simple Discontinuous Mesh Evolution

TL;DR

This paper introduces a space-time cut finite element method for the heat equation on overlapping meshes with discontinuous evolution, enabling stable and accurate simulations with optimal error estimates.

Contribution

It develops a novel finite element formulation using Nitsche's method for evolving overlapping meshes with discontinuous changes, extending existing analysis techniques.

Findings

Achieves optimal order a priori error estimates.

Verifies convergence orders through numerical experiments.

Handles discontinuous mesh evolution with stability and accuracy.

Abstract

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/"on top" of it. Here the overlapping mesh is prescribed a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson in [1, 2]. The greatest modification is the introduction of a Ritzlike "shift operator" that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.