For many application problems in the natural and technical sciences the behavior of macroscopicquantities strongly depends on small time-dependent influences and the structure of the underlyingmedium. The large variability of heterogeneities on different spatial and temporal scales seemsto be hard to capture using purely deterministic mathematical models. Specific examples ofsuch situations are the solute transport in complex geological formations, macroscopic transportphenomena in tumors, phase transformation processes in materials science, or the dynamics ofatmospheric flows. From the viewpoint of modeling a promising alternative is the use of stochasticevolution equations. In the last two decades there has been substantial progress in theory andnumerics for probabilistic partial differential equations, in particular for linear elliptic equations.However, for nonlinear and/or convection dominated problems the field is still in its infancy.In this project we plan to develop, implement, and analyze higher-order Discontinuous-Galerkin methods(DG) for a class of nonlinear convection-dominated evolution equations including randomprocesses in time and space. Time-discretizations for Wiener processes appropriate for high-orderspace discretization will be constructed. To reduce the extreme computational expensesassociated with spatial noise when employing classical Monte-Carlo-methods (MC) we shall usethe Karhunen-Loeve based moment-equation approach (KLME) for the spatial noise. This has tobe integrated in the framework of DG-methods for the evolution equation itself. Technically wewant to construct a dynamical a-posteriori error control that governs the mesh parameter, theorder of the DG-method, and the finite dimension of the stochastic parameterization.The overall goal on the numerical side is to apply the algorithm to a number of realistic subsurfaceflow problems in a heterogeneous randomly given geological formation. Concerning analysis weseek to obtain well-posedness results for simple model problems and first a-priori/a-posteriori errorestimates for the used approximation methods.
09/2008 - 08/2011
SRC SimTech / Deutsche Forschungsgemeinschaft