Discretizations of nonlinear parabolic equations

The project concerns the numerical analysis of nonlinear parabolic equations, including the evolution of the p- Laplacian, which are found in technical applications ranging from such diverse areas as diffusion in porous media and image processing. Obtaining a theoretical understanding for the numerical approximations of such problems is crucial, e.g., when designing efficient modelling software. We have previously focused on the analysis of time discretizations of these equations and in my thesis "Discretizations of nonlinear dissipative evolution equations" a fully nonlinear convergence analysis was developed. The results of the analysis include the very same convergence orders for algebraically stable Runge-Kutta methods and A-stable multistep methods as found when discretizing stiff ODEs. This is a significant improvement of the previously published convergence results, which are either to pessimistic or not valid for genuinely nonlinear problems. The goal of the project is to extend the analysis even further as well as linking it to spatial discretizations like the Galerkin method and finite differences. The methodology of the project is to merge our time discretization framework with the very successful one presented by the group around Prof. Alexander Ostermann and combining this with the transverse method of lines. Preliminary studies of the full Runge-Kutta/Galerkin discretizations indicate that the analysis has great potential. Although the project emphasizes the development of new theoretical results, it also includes applications in medicine and engineering.