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

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 O(NlogN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}(N\log N)$\end{document} degrees of freedom are required, compared to O(N2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}(N^{2})$\end{document} 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 O(N3/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}(N^{3/2})$\end{document} degrees of freedom. Numerical results are provided that demonstrate the sharpness of the estimates and the efficiency of the method.