Numerical Algorithms

, Volume 78, Issue 3, pp 929–956 | Cite as

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

  • Rainer HartmannEmail author
  • Christoph Pflaum
Original Paper


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.


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


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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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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Computer Science, System SimulationFriedrich-Alexander-Universität Erlangen-Nürnberg (FAU)ErlangenGermany

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