PhD Defence: A Non-standard Finite Element Method using Boundary Integral Operators

A Non-standard Finite Element Method using Boundary Integral Operators

The thesis is concerned with a non-standard finite element method, referred to as a BEM-based FEM, which is based on element-local boundary integral operators and which permits polyhedral element shapes as well as meshes with hanging nodes. The method employs elementwise PDE-harmonic trial functions and can thus be interpreted as a local Trefftz method. The construction principle requires the explicit knowledge of the fundamental solution of the partial differential operator, but only locally, i.e., in every polyhedral element. This allows us to solve PDEs with elementwise constant coefficients. Some major results of the thesis are presented. Among these are rigorous error estimates of optimal order for a model problem, construction and convergence rates of a fast solution technique based on the ideas of the one-level FETI approach, and the applicationof the method to convection-diffusion problems where we draw parallels to established stabilization techniques. Some numerical examples confirm the theoretical results.