hp-FEM for large-scale singular three-dimensional problems

Higher order finite element methods are well-established tools for the analysis of partial differential equations that arise from physical, engineering, or mathematical problems. They prove to be very efficient compared to standard finite element methods, yielding unconditional exponential convergence if used appropriately. They also are suitable for problems exhibiting multiscale behavior or containing singularities. However, there are some challenging problems that impede wider use of the method, e.g., in commercial software. The main aim of this work is to address these problems. In particular, it is well-known that the linear problems yielded by the hp-FEM are usually much smaller than the ones produced by standard methods. Unfortunately, the problem also becomes very ill-conditioned and sometimes even causes the iterative solvers to fail. We present several ways how to deal with this issue and define new sets of shape functions with better conditioning properties. An essential part of this work was to develop a proof-of-concept software allowing us to verify all the aforementioned ideas. During the work on the HERMES project we created an hp-FEM package capable of solving elliptic problems on unstructured tetrahedral meshes. We have encountered several interesting theoretical and algorithmic issues while implementing the software. We discuss them in what follows, and present a brief comparison to the 2D case. We present a short description of the software together with some test cases and numerical examples. (Abstract shortened by UMI.