Robust Numerical Methods for partial Differential Equations

During the last decades numerical simulations has become an indispenisble tool in science and engineering. In particular numerical approximations of partial differential equations are frequently an important ingredient in the design of such programs. Nume rical approx. of partial differential equations aer used in various areas like structure mechanics molecular biology, weather forecasts, reservoir simulations, climate predictions, simulation of colliding black holes and so on. Even if the most basic the ory of numerical methods for partial differential is well understood, there are still many challenging problems in this area. The technological development of modern computers has made it possible to consider more complex models and algorithms With improv ed memory and speed requirements. However, in order to understand these complex processes, new challenging mathematical problems arise. In the present project we will study the construction of robust numerical methods for systems of partial differential equations. Hnce, we are studying systems which depend critically on certain parameters, and our goal is to construct numerical methods which performs well uniformly with respect to these parameters. This general approach will be applied to model problems arising in areas like fluid flow, porous medium flow, alloy solidification, phase transition and so on.