New approximation algorithms for a class of partial differential equations displaying boundary layer behavior

The aim of this article is to propose and study a class of new algorithms which qualitatively and quantitatively capture the behavior of the exact solutions of a class of evolution partial differential equations which display boundary layer behavior. The idea of the new schemes is to incorporate the boundary layer into the Galerkin base in the finite element approximation. Our error estimates demonstrate that the new schemes are effective in the under-resolved region of the classical schemes. Our numerical experiments support the numerical analysis. The design and analysis of the new schemes depend on the detailed analysis of the boundary layer. The development and proof of the asymptotic expansion of the solutions of the partial differential equations are attached as an Appendix.