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

稀疏网格是一种减少偏微分方程组数值解计算量的创新技术。应用于具有可重入边和角的复杂区域上的微分方程或高维问题，如与时间无关的薛定谔方程。在这两种情况下，都很难获得准确的数值解。为了将稀疏网格应用于此类微分方程，采用Ritz-Galerkin离散化是很重要的。然而，在变系数的情况下，这样的离散化导致了几个算法困难。对于最近发展起来的稀疏网格上的一种新的离散化方法来说，这些困难没有出现。这种离散化应用了预小波和具有半正交性的离散化。利用这一概念，可以用适当的算法有效地求解变系数偏微分方程组。该项目的目标是继续开发求解稀疏网格上的偏微分方程组的算法。特别是，必须开发用于变系数的自适应稀疏网格的算法和计算刚度矩阵的高效算法。此外，由于传统的并行化概念不能应用于稀疏网格，因此需要新的并行化概念。新算法将被实现，并针对合适的应用进行分析。

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.