Citation/Abstract

The hp -FEM is a modern version of the Finite Element Method (FEM) which admits elements with different sizes ( h ) and different polynomial degrees ( p ) to obtain extremely fast (exponential) convergence rates. We improve the performance of the hp -FEM by using a new basis respecting the De Rham diagram. Also, we approximate curved boundaries by NURBS curves instead of the commonly used line segments, which alleviates variational crimes. The main goal of the thesis is automatic adaptivity on higher-order edge elements. Higher-order elements can be refined in a variety of ways--the polynomial degree of an element can be increased without spatial refinement, or the element can be split into four or two sons with various combinations of polynomial degrees in the sons. We develop a new hp -adaptive strategy based on arbitrary-level hanging nodes with optimal selection of element refinements. The basic idea is that forced (unwanted) refinements are avoided, thus the complexity of the hp -adaptive algorithm reduces while the efficiency improves. The technique is illustrated on several examples.