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  unknowns. The corresponding discretization error is  in the -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 . 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 -norm is  for  and  for .

Researchers should cite this work as follows:

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

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1. differential equations
2. discretization
3. Illinois
4. NanoBio Node at Illinois
5. prewavelets
6. sparse matrix