ON A MONOTONIC CONVECTION-DIFFUSION SCHEME IN ADAPTIVE MESHES

In this article we apply our recently proposed upwind model to solve t he two-dimensional steady convection-diffusion equation in adaptive me shes. In an attempt to resolve high-gradient solutions in the flow, we construct finite-element spaces through use of Legendre polynomials. According to the fundamental analysis conducted in this article, we co nfirm that this finite-element model accommodates the monotonicity pro perty. According to M-matrix theory, we know within what range of Pecl et numbers the Petrov-Galerkin method can perform well in a sense that oscillatory solutions are not present in the flow. This monotonic reg ion is fairly restricted, however, and limits the finite-element pract ioner's choices of a fairly small grid size. This limitation forbids a pplication to practical flow simulations because monotonic solutions a re prohibitively expensive to compute. Circumvention of this shortcomi ng is accomplished by remeshing the domain in an adaptive way. To alle viate the excessive memory requirement, our implementation incorporate s a reverse Cuthill-McKee (RCM) renumbering technique. Numerical resul ts are presented in support of the ability of the finite-element model developed herein to resolve sharp gradients in the solution. Also sho wn from these numerical exercises is that considerable savings in comp uter storage and execution time are achieved in adaptive meshes throug h use of the RCM element-reordering technique.