On a monotonic convection-diffusion scheme in adaptive meshes

In this article we apply our recently proposed upwind model to solve the two-dimensional steady convection-diffusion equation in adaptive meshes. 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 confirm that this finite-element model accommodates the monotonicity property. According to M-matrix theory, we know within what range of Peclet numbers the Petrov-Galerkin method can perform well in a sense that oscillatory solutions are not present in the flow. This monotonic region is fairly restricted, however, and limits the finite-element practioner's choices of a fairly small grid size. This limitation forbids application to practical flow simulations because monotonic solutions are prohibitively expensive to compute. Circumvention of this shortcoming is accomplished by remeshing the domain in an adaptive way. To alleviate the excessive memory requirement, our implementation incorporates a reverse Cuthill-McKee (RCM) renumbering technique. Numerical results are presented in support of the ability of the finite-element model developed herein to resolve sharp gradients in the solution. Also shown from these numerical exercises is that considerable savings in computer storage and execution time are achieved in adaptive meshes through use of the RCM element-reordering technique.