Abstract：

The linear reaction-diffusion problem $-\varepsilon^2\Delta u + b u = f$ is considered on the unit square with homogeneous Dirichlet boundary conditions. Here $\varepsilon$ 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} + \varepsilon^{1/2} N^{-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(N^{2})$ to $ O(N^{3/2})$.

报告人简介：Martin Stynes 教授的主要研究领域是奇异摄动微分方程的数值解法，包括对流扩散方程，线性化Navier-Stokes方程等。