A Non-standard Finite Element Method using Boundary Integral Operators

Dr. Clemens Hofreither

Dec. 18, 2012, 1 p.m. S3 056

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.