A cut finite element method for the heat equation on overlapping meshes: Energy analysis for cG(1) mesh movement

We present a cut finite element method for the heat equation on two overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a stationary background mesh at the bottom and an overlapping mesh that is allowed to move around on top of the background mesh. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing or evolving interior geometry. In this paper the overlapping mesh is prescribed a cG(1) movement, 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 that mimics the standard discontinuous Galerkin time-jump term. The cG(1) mesh movement results in a space-time discretization for which existing analysis methodologies either fail or are unsuitable. We therefore propose, to the best of our knowledge, a 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. $*$ UPDATE and CORRECTION: After this work was made public, it was discovered that the core components of the new energy analysis framework seemed to have been discovered independently by us and Cangiani, Dong, and Georgoulis in [1].