Efficiency and Accuracy of Higher-Order Boundary-Element Methods for Steady Convective Heat Diffusion

Abstract

Higher-order boundary-element methods (BEM) are presented for steady-state convective diffusion problems in two dimensions. The free-space steady convective diffusion fundamental solutions considered in this article provide an analytical upwinding for the entire Peclet number range, from zero to infinity. However, integration of the kernels over the boundary elements requires considerable attention, especially at higher Peclet numbers. We define an influence domain due to these convective kernels and then localize the surface integrations only within the domain of influence. The localization of the kernels becomes more prominent as the Peclet number of the flow increases. This, in turn, leads to increasing sparsity and improved conditioning of the global matrix. Consequently, iterative solvers become the primary choice. We consider an example problem with an exact solution, and investigate the accuracy and efficiency of the higher-order BEM formulations for Peclet numbers in the range from 200 to 200,000. The quartic boundary elements included in this study are shown to be extremely efficient.