On Parabolic Boundary Layers for Convection–Diffusion Equations in a Channel: Analysis and Numerical Applications

In this article we discuss singularly perturbed convection–diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, when this structure is available; see e.g. [Cheng, W. and Temam, R. (2002). Comput. Fluid. V.31, 453–466; Jung, C. (2005). Numer. Meth. Partial Differ. Eq. V.21, 623–648]. This approach is developed in this article for a convection–diffusion equation. Using an analytical approach, we first derive an approximate (simplified) form of the parabolic boundary layers (elements) for our problem; we then develop new numerical schemes using these boundary layer elements. The results are performed for the perturbation parameter ε in the range 10−1–10−15 whereas the discretization mesh is in the range of order 1/10–1/100 in the x-direction and of order 1/10–1/30 in the y-direction. Indications on various extensions of this work are briefly described at the end of the Introduction.