A new stable method for singularly perturbed convection–diffusion equations

Abstract

We present a new approach in the finite element framework to solve singularly perturbed convection–diffusion problems. Standard methods for these problems perform poorly because of the presence of sharp boundary and/or internal layers in the solution. The operator is elliptic within these layers and essentially hyperbolic away from them. In the proposed approach, the test function space is modified and hence fits into the class of Petrov–Galerkin finite element methods. The proposed finite element approach is motivated by, and is similar to, one proposed by Hemker [P.W. Hemker, A numerical study of stiff two-point boundary problems, Mathematical Center Tracts, vol. 80, Mathematisch Centrum, Amsterdam, 1977] for one-dimensional problems which was based on using Green’s functions to motivate appropriate test spaces. For problems in higher dimensions we no longer utilize Green’s functions, but rather we try to capture the solutions of suitably posed discretized dual problems with functions in the test space. The dual problem is posed in such a way that a weighted error at the interelement boundaries is minimized. The dual solution is exponential. Hence, the functions in the test space are exponentially fitted to provide a better approximation to the dual solution. These modified test functions fit into a hierarchical framework resulting in the development of a stable higher-order method. Accurate numerical solutions are obtained on structured and unstructured meshes, and numerical artifacts are confined to the few elements containing the boundary layers.