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Numerical Algorithms

, Volume 78, Issue 3, pp 929–956

# A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids

• Rainer Hartmann
• Christoph Pflaum
Original Paper

## Abstract

We present a Ritz-Galerkin discretization on sparse grids using prewavelets, which allows us to solve elliptic differential equations with variable coefficients for dimensions d ≥ 2. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This algorithm is based on standard 1-dimensional restrictions and prolongations, a simple prewavelet stencil, and the classical operator-dependent stencil for multilinear finite elements. Numerical simulation results are presented for a three-dimensional problem on a curvilinear bounded domain and for a six-dimensional problem with variable coefficients. Simulation results show a convergence of the discretization according to the approximation properties of the finite element space. The condition number of the stiffness matrix can be bounded below 10 using a standard diagonal preconditioner.

## Keywords

Sparse grid Prewavelets Semi-orthogonality Variable coefficients Conjugate gradient method Finite element method

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## Notes

### Acknowledgments

The authors gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG) in the framework of the German excellence initiative.

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## Copyright information

© Springer Science+Business Media, LLC 2017

## Authors and Affiliations

• Rainer Hartmann
• 1
• Christoph Pflaum
• 1
1. 1.Department of Computer Science, System SimulationFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany