===affil2:
Friedrich-Alexander-Universität Erlangen-Nürnberg 
===firstname:
Christoph
===firstname4:

===firstname3:

===lastname2:
Hartmann
===keyword1:
Discretization
===lastname:
Pflaum
===firstname5:

===affil6:

===lastname3:

===email:
christoph.pflaum@fau.de
===keyword2:
Multigrid
===keyword_other2:

===lastname6:

===affil5:

===otherauths:

===lastname4:

===affil4:

===lastname7:

===competition:
no
===affil7:

===firstname7:

===postal:
Friedrich-Alexander-Universität Erlangen-Nürnberg 
Department Computer Science 10
Cauerstraße 11 
D-91058 Erlangen
===firstname6:

===ABSTRACT:
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 $O(N (\log N)^{d-1})$ unknowns. The corresponding discretization error is $O(N^{-1} (\log N)^{d-1})$ 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 $O(N^{-1} (\log N)^{d-1})$ for $d \leq 4$ and $O(N^{-(2/(d-2))})$ for $d \geq 5$.
===affil3:

===keyword_other1:
Sparse Grids
===lastname5:

===affilother:

===title:
Discretization of Elliptic Differential Equations Using Sparse Grids and Prewavelets
===firstname2:
Rainer