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A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions
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2008
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Numerical AnalysisFinite Element MethodGalerkin Finite-element MethodMethod Of Fundamental SolutionEngineeringReaction-diffusion ProblemSingularly Perturbed ProblemPerturbation MethodNumerical ComputationUnit SquareSemi-implicit MethodNumerical SimulationAnomalous DiffusionBoundary Element MethodNumerical MethodsAssociated Energy NormNumerical Method For Partial Differential EquationMultiscale Modeling
The linear reaction–diffusion problem – ϵ2Δu + bu = f is considered on the unit square with homogeneous Dirichlet boundary conditions. Here ϵ 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 + ϵ1/2N−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.