A cut finite element method for the heat equation on overlapping meshes: L^2-analysis for dG(0) mesh movement
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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 dG(0) movement, meaning that its location as a function of time is 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 dG(0) mesh movement results in a space-time discretization with a nice 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 [12, 13], here referred to as an L^2-analysis because of the norm used in the error analysis. The greatest modification is the use of a shift operator that generalizes the Ritz projection operator. The shift operator is used to handle the shift in the overlapping mesh's location at discrete times. The L^2-analysis consists of the corresponding standard stability estimates and 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.
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