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Advances in Computational Mathematics

, Volume 41, Issue 6, pp 987–1014 | Cite as

A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem

  • Niall MaddenEmail author
  • Stephen Russell
Article

Abstract

We consider the numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem posed on the unit square by a multiscale sparse grid finite element method. A Shishkin mesh which resolves the boundary and corner layers, and yields a parameter robust solution, is used. Our analysis shows that the method achieves essentially the same level of accuracy, in the energy norm, as the standard Galerkin finite element method with bilinear elements. However, only \(\mathcal {O}(N\log N)\) degrees of freedom are required, compared to \(\mathcal {O}(N^{2})\) for the corresponding Galerkin finite element method. This may be regarded as a generalisation of Liu et al. (IMA J. Numer. Anal. 29(4), 986–1007 2009) which used a two-scale method requiring \(\mathcal {O}(N^{3/2})\) degrees of freedom. Numerical results are provided that demonstrate the sharpness of the estimates and the efficiency of the method.

Keywords

Singularly perturbed Reaction-diffusion Shishkin mesh Sparse grid 

Mathematics Subject Classifications (2010)

65N15 65N30 65Y20 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

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