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

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+ε^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(N^3/2). These results are then generalized to systems of reaction-diffusion equations.