Cut Finite Element Methods on Overlapping Meshes: Analysis and Applications

Abstract: This thesis deals with both analysis and applications of cut finite element methods (CutFEMs) on overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a background mesh at the bottom and one or more overlapping meshes that are stacked on top of it. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing geometry. The main content of the thesis is the five appended papers. The thesis consists of an analysis part and an applications part. In the analysis part (Paper I and Paper II), we consider cut finite element methods on overlapping meshes for a time-dependent  parabolic model problem: the heat equation on two overlapping meshes, where one mesh is allowed to move around on top of the other. In Paper I, the overlapping mesh is prescribed a cG(1) movement, meaning that its location as a function of time is continuous and piecewise linear . 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 enough to be applicable to the current setting. In Paper II, the overlapping mesh is prescribed a dG(0) movement, meaning that its location as a function of time is  discontinuous  and  piecewise constant . The dG(0) mesh movement results in a space-time discretization for which existing analysis methodologies work with some modifications to handle the shift in the overlapping mesh's location at discrete times. The applications part (Paper III, IV, and V) concerns cut finite element methods on overlapping meshes for  stationary  PDE-problems. We consider two potential applications for CutFEM on overlapping meshes. The first application, presented in Paper III, presents methodology for evaluating configurations of buildings based on wind and view. The wind model is based on a CutFEM on overlapping meshes for Stokes equations. The second application, presented in Paper IV and Paper V, concerns a software application (app). The app lets a user define and solve physical problems governed by PDEs in an immersive and interactive augmented reality environment.

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