An important tool in engineering is the finite element method.... The combination of both methods, the h-p version, supplies the pre-asymptotic exponentially convergent p-version continuously with properly adapted grids. Hence it achieves the superior exponential convergence asymptotically, too, instead of algebraic convergence of its ingredients the h-version and the p-version. Although the first theoretical results claiming these convergence rates are quite classic, the number of codes using the h-p-version of finite elements is still rather limited. Reasons for that are the pure implementational complexity and the details, in conjunction with the rumor of engineers' low precision requirements. But the major reason is the lack of a robust (self-) adaptive control delivering the desired exponential convergence. ...
In the this thesis we present some steps towards an efficient implementation of the theoretically known exponential convergence. As it turns out, an efficient implementation requires additional theoretical considerations, which play a major role there as well. This includes both the fully automatic h-p-version and as a subset the p-version on suitable grids. We present some details concerning our approach implementing an adaptive h-p-version based on an adaptive multilevel h-version code named Kaskade. This software package uses unstructured grids of triangles in two dimensions and tetrahedra in three dimensions.