Procedures in Adaptive Finite Element Analysis
The present work is concerned with topics related to some adaptive methods for the approximate solution of differential equations. Emphasis is on the finite element method for second order elliptic equations, where we use either p-refinement or remeshing with an advancing front mesh generator in order to improve approximate solutions. However, we also show how the concept of hierarchical p-refinement may be used in numerical integration of systems of first order differential equations, that arise e.g. in semidiscrete methods for parabolic problems.
A brief outline of the adaptive concept is presented, and in particular we focus on the hierarchical p-version of the finite element method. The numerical stability of p-type equation systems is investigated and we suggest some solution strategies for the sequence of nested equation systems that arises from an adaptive hierarchical formulation. Nested iteration, multigrid methods and preconditioned conjugate gradient methods are described in short, as the advocated solution schemes are based on these concepts.
Tetrahedral mesh generation by the so-called advancing front technique is discussed. The algorithm is described, some remarks on execution times are made, and we elaborate on the possibility to use this method for adaptive remeshing. Robustness is a prime concern, particularly so in an adaptive context, so the sensitivity of the algorithm with respect to its various variables has been investigated. A couple of methods that enhance the reliability are also described. Furthermore, we develop an algorithm that creates exceptionally stretched elements adjacent to domain boundaries. The suggested method combines in a natural way with the advancing front concept.
Finally, numerical integration of stiff first order systems of differential equations is treated. We show how a C0-continuous polynomial time discretization, in conjunction with a weighted residual method, can be used to derive methods that correspond to the diagonal and first subdiagonal Padé approximants. A hierarchical basis is used to retrieve integration schemes of ascending order. The approach is analogous to the p-version of the finite element method for elliptic problems, so it allows for local refinements in an obvious manner. In addition, it becomes straightforward to perform error estimation by an embedding technique, and we utilize this idea to design an adaptive method for the semidiscrete system.