Discretization of Elliptic Differential Equations Using Sparse Grids and Prewavelets

By Christoph Pflaum

Friedrich-Alexander-Universität Erlangen-Nürnberg

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Sparse grids can be used to discretize second order elliptic differential equations on a d-dimensional cube. Using Galerkin discretization, we obtain a linear equation system with $ O(N (%5Clog N)^{d-1})$ unknowns. The corresponding discretization error is $ O(N^{-1} (%5Clog N)^{d-1})$ in the $ H^1$-norm. A major difficulty in using this sparse grid discretization is complexity of the related stiffness matrix. Consequently, only differential equations with constant coefficients could be efficiently discretized using sparse grids for $ d>2$. To reduce the complexity of the sparse grid discretization matrix, we apply pre-wavelets. This simplifies the implementation of the multigrid Q-cylce. Furthermore, we present a new sparse grid discretization for Helmholtz equation with a variable coefficient c. This discretization utilizes a semi-orthogonality property. The convergence rate of this discretization in $ H^1$-norm is $ O(N^{-1} (%5Clog N)^{d-1})$ for $ d %5Cleq 4$ and $ O(N^{-(2/(d-2))})$ for $ d
%5Cgeq 5$.

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Researchers should cite this work as follows:

  • Christoph Pflaum (2016), "Discretization of Elliptic Differential Equations Using Sparse Grids and Prewavelets," http://nanohub.org/resources/23520.

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