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

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.

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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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Correspondence to Rainer Hartmann.

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Hartmann, R., Pflaum, C. A prewavelet-based algorithm for the solution of second-order elliptic differential equations with variable coefficients on sparse grids. Numer Algor 78, 929–956 (2018). https://doi.org/10.1007/s11075-017-0407-9

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Keywords

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