Adaptive time-space discretizations in 4 examples.

Despite of the use of supercomputers, nonlinear time-dependent problems are subject to current research in computational mathematics. To succeed in numerical treatment, a suitable discretization in time and space is required, but is complicated by the fact that the discretizations of both dimensions are mutually dependent. With phase transitions, contact problems and also American options singularities and/or free boundaries appear. Therefore, highly adapted meshes in some parts of the space-time area are needed, but are unknown a priori. The main attention of this project lies on the development of suitable discretizations and algorithms which take such phenomena into consideration. Four model problems describe for basic conditions the typical behaviour of solutions in four innovative application fields: satellite dynamics, phases field equations, quantum mechanics and computational finance. Each of these four mathematical models can be treated in a similar analytical context and should be tested empirically in numerical experiments. The purpose of this project is to develop more efficient algorithms which allow simulations, which are becoming more and more complicated for new and innovative application fields.