Adaptive and parallel algorithms for solving partialdifferential equations with variable coefficients on sparse grids

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Sparse grids are an innovative technique for reducing the computational amount for the numerical solution of partial differential equations. Applications are differential equations on complex domains with reentrant edges and corners or high dimensional problems like the time independent Schrödinger equation. In both cases, accurate numerical solutions are difficult to obtain. In order to apply sparse grids to such differential equations it is important to apply a Ritz-Galerkin discretization. However, such a discretization leads to several algorithmic difficulties in case of variable coefficients. These difficulties do not appear for a new discretization method on sparse grids, which was recently developed. This discretization applies prewavelets and a discretization with semi-orthogonality. By this concept, PDE’s with variable coefficients can efficiently be solved by suitable algorithms. The aim of the project is to continue the development of algorithms for solving PDE’s on sparse grids. In particular algorithms on adaptive sparse grids for variable coefficients and efficient algorithms for the calculation of the stiffness matrix have to be developed. Furthermore, new parallelization concepts are needed, since conventional parallelization concepts cannot be applied to sparse grids. The new algorithms will be implemented and analyzed for suitable applications.