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.