2023/02/01 by Mats G. Larson, Anders Logg, Larson, Mats G. +3
Engineering · Mathematics · #65M12 #65M22 #65M60 #65M85 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2302.00337
openalex publication_date 2023/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.