Generalized mixed finite element methods: cut elements and virtual elements

QC 2024-10-07

Abstract

A multitude of physical phenomena are accurately modeled by partial differential equations (PDEs). These equations are complicated to solve in general, and when an analytical solution is not able to be found, a numerical method can give an approximate solution. This can be very useful in many applications. This thesis explores the development and analysis of cut finite element methods (CutFEM) for discretising PDEs with a focus on preserving divergence conditions essential in applications such as fluid dynamics and electromagnetism. CutFEM has been developed with the aim to simplify distretising PDEs in domains with complicated geometries, by allowing the geometry to be positioned arbitrarily relative to the computational mesh. Traditional CutFEM have failed to maintain the divergence conditions, leading to numerical inaccuracies. Following the mixed finite element method (FEM) framework, the research contained herein introduces novel strategies that preserve the divergence at the discrete level and addresses other key challenges when discretizing PDEs in geometries unfitted to the computational mesh. For example, the techniques are also able to control the condition number of the linear systems. The virtual element method (VEM) is another method able to handle complicated geometries. It does this by allowing for a mesh to be constructed from general polytopal elements, not just triangles or rectangles. One work of the thesis investigates the spectral condition number of the mixed VEM, demonstrating the effectiveness of auxiliary space preconditioning in bounding spectral condition numbers independently of mesh element aspect ratios.

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