The main topic of this thesis is a new method for grid deformation with applications in r- and rh-adaptive methods. The new grid deformation method is analysed theoretically and numerically.

The thesis consists of six chapters. After the introduction, I introduce the basic grid deformation algorithm combined with a mathematical analysis of its main properties. My method is a generalisation of Liao`s method with enhanced flexibility and robustness. The third chapter contains a new convergence theory for grid deformation. The statements made are mathematically proven and confirmed by numerical experiments. By sophisticated iterations of the basic grid deformation method, I construct new deformation algorithms with improved accuracy and robustness. By exploiting the given grid hierarchy, I develop a multilevel variant of my deformation method, which features both optimal asymptotic complexity and optimal convergence order.
The fourth chapter deals with applying grid deformation to the computation of the Poisson equation on the L-domain. By grid deformation, one can recover the optimal order of convergence with respect to the gradient norm. The grid deformation process is controlled entirely by a posteriori error estimation and does not need explicit input by the user.
In the fifth chapter, I compare r-adaptivity with common h-adaptive strategies. It turns out that by combination of these approaches (rh-adaptivity), a major acceleration compared to r-adaptivity can be achieved without sacrifying accuracy.
The last chapter contains the application of the rh-adaptive algorithm developed so far to a generalised Poisson equation with an anisotropic diffusion tensor. Using the DWR-method, the error of derived quantities like the error in a single point is reduced significantly.