Space-Time CutFEM on Overlapping Meshes: Simple Continuous Mesh Motion

TL;DR

This paper introduces a novel cut finite element method for the heat equation on overlapping meshes with simple continuous motion, employing Nitsche's method and a new energy analysis framework to ensure stability and optimal error estimates.

Contribution

It develops a space-time discretization approach for moving overlapping meshes using continuous Galerkin and discontinuous Galerkin methods, with a new energy analysis framework for stability and error estimation.

Findings

Achieves optimal order error estimates in numerical experiments.

Verifies stability and convergence through one-dimensional numerical results.

Employs a new energy analysis framework suitable for the complex mesh motion setting.

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 moves around inside/"on top" of it. Here the overlapping mesh is prescribed a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. 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 and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively new energy analysis framework that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and 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.