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IMA Journal of Numerical Analysis Advance Access originally published online on October 3, 2008
IMA Journal of Numerical Analysis 2009 29(4):986-1007; doi:10.1093/imanum/drn048
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A two-scale sparse grid method for a singularly perturbed reaction–diffusion problem in two dimensions

Fang Liu{dagger}

School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081, China

Niall Madden{ddagger}

Department of Mathematics, National University of Ireland, Galway, Ireland

Martin Stynes§

Department of Mathematics, National University of Ireland, Cork, Ireland

Aihui Zhou

The State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

{dagger} Email: fliu{at}lsec.cc.ac.cn

{ddagger} Email: niall.madden{at}nuigalway.ie

§ Corresponding author. Email: m.stynes{at}ucc.ie

Email: azhou{at}lsec.cc.ac.cn

Received on 12 November 2007. Revised on 10 May 2008.


  Abstract

The linear reaction–diffusion problem – {epsilon}2{Delta}u + bu = f is considered on the unit square with homogeneous Dirichlet boundary conditions. Here {epsilon} is a small positive parameter and the problem is in general singularly perturbed. The numerical solution of this problem is analysed on a Shishkin mesh that has N intervals in each coordinate direction, using the Galerkin finite-element method with bilinear trial functions. The accuracy of this method, measured in the associated energy norm, is shown to be O(N–2 + {epsilon}1/2N–1 ln N). It is proved that a two-scale sparse grid method achieves the same order of accuracy while reducing the number of degrees of freedom from O(N2) to O(N3/2). These results are then generalized to systems of reaction–diffusion equations.

Key Words: reaction–diffusion; finite element; Shishkin mesh; two-scale discretization; sparse grid


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