LAYER STRUCTURE AND THE GALERKIN FINITE ELEMENT METHOD FOR A SYSTEM OF WEAKLY COUPLED SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS WITH MULTIPLE SCALES

We consider a system of weakly coupled singularly perturbed convection-diffusion equations with multiple scales. Based on sharp estimates for first order derivatives, Linss [T. Linss, Computing 79 (2007) 23-32.] analyzed the upwind finite-difference method on a Shishkin mesh. We derive such sharp bounds for second order derivatives which show that the coupling generates additional weak layers. Finally, we prove the first robust convergence result for the Galerkin finite element method for this class of problems on modified Shishkin meshes introducing a mesh grading to cope with the weak layers. Numerical experiments support our theory.