Xiangyun Meng, Associate Professor, School of Mathematics and Statistics, Beijing Jiaotong University

Shuo Zhang, Associate Professor

Balanced-norm and energy-norm error analyses for a backward Euler/FEM solving a singularly perturbed parabolic reaction-diffusion problem

10:00-11:00 October 26(Wednesday), Z305

In the derivation of error bounds, uniformly in the singular perturbation parameter, for finite element methods (FEMs) applied to elliptic singularly perturbed linear reaction-diffusion problems, the usual energy norm is unsatisfactory since it is essentially no stronger than the L2 norm. Consequently several researchers have analysed errors in FEM solutions, uniformly in the singular perturbation parameter, using balanced norms whose H1 component is weighted correctly to maintain its influence. But the derivation of energy and balanced-norm error bounds for FEM solutions of singularly perturbed reaction-diffusion problems is confined almost entirely to steady-state elliptic problems — little has been proved for time-dependent parabolic singularly perturbed problems. This talk addresses this gap in the literature: the backward Euler method in time, combined with a bilinear FEM on a spatial Shishkin mesh, is applied to solve a parabolic singularly perturbed reaction-diffusion problem, and energy-norm and balanced-norm error estimates, which are uniform in the singular perturbation parameter, are derived — these results are stronger than any previous results of the same type. Furthermore, numerical experiments demonstrate the sharpness of our error bounds.