Space-Time CutFEM on Overlapping Meshes: Simple Continuous Mesh Motion
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.
Cite
@article{arxiv.2302.00337,
title = {Space-Time CutFEM on Overlapping Meshes: Simple Continuous Mesh Motion},
author = {Mats G. Larson and Anders Logg and Carl Lundholm},
journal= {arXiv preprint arXiv:2302.00337},
year = {2023}
}
Comments
Preprint for journal submission. arXiv admin note: text overlap with arXiv:2301.13517