Applications of Special Functions in High Order Finite Element Methods

Abstract

In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given.