Numerical experiments for advection-diffusion problems in a channel with a 180° bend

In this paper we solve numerically two singularly perturbed linear convection±diffusion problems for heat transfer in a ̄uid with an assumed ̄ow ®eld in the neighbourhood of a 180° bend in a channel. In the ®rst problem the theoretical solution has a parabolic boundary layer and in the second problem there is both a parabolic and a regular boundary layer in the solution. The numerical method uses piecewise uniform ®tted meshes condensing in a neighbourhood of each boundary layer and a standard upwind ®nite di€erence operator satisfying a discrete maximum principle. The numerical results con®rm computationally that the method is e-uniform in the sense that the rate of convergence and the error constant of the method are independent of the singular perturbation parameter e, where e denotes the reciprocal of the P eclet number of the ̄uid. This e-uniform behaviour is obtained only when an appropriate piecewise uniform ®tted mesh is constructed for each boundary layer. This is con®rmed by several additional computations on meshes which do not ful®ll this requirement. Ó 2001 Elsevier Science Inc. All rights reserved.