SUMMARYA boundary element method (BEM) for steady viscous #uid #ow at high Reynolds numbers is presented. The new integral formulation with a poly-region approach involves the use of the convective kernel with slight compressibility that was previously employed by Grigoriev and Fafurin [1] for driven cavity #ows with Reynolds numbers up to 1000. In order to avoid the overdeterminancy of the global set of equations when using eight-noded rectangular volume cells from that previous work, 12-noded hexagonal volume regions are introduced. As a result, the number of linearly independent integral equations for each node becomes equal to the degrees of freedom of the node. The numerical results for square-driven cavity #ow having Reynolds numbers up to 5000 are compared to those obtained by Ghia et al.[2] and demonstrate a high level of accuracy even in resolving the secondary vortices at the corners of the cavity. Next, a comprehensive study is done for backward-facing step #ows at Re"500 and 800 using the BEM, along with a standard Galerkin-based "nite element methods (FEM). The numerical methods are in excellent agreement with the benchmark solution published by Gartling [3]. However, several additional aspects of the problem are also considered, including the e!ect of the in#ow boundary location and the traction singularity at the step corner. Furthermore, a preliminary comparative study of the poly-region BEM versus the standard FEM indicates that the new method is more than competitive in terms of accuracy and e$ciency.

SUMMARYDespite the signiÿcant number of publications on boundary element methods (BEM) for time-dependent problems of heat di usion, there still remain issues that need to be addressed, most importantly accuracy of the numerical modelling. Although very precise for steady-state problems, the common boundary element methods applied to transient problems do not yield highly accurate numerical solutions. This paper investigates the reasons that prohibit achievement of a high level of accuracy for transient heat di usion problems with continuous temperature and bounded heat ux solutions. In order to greatly enhance the commonly used boundary element formulations, we propose higher-order time interpolation functions, including quadratic and quartic approximations. We show that the use of higher-order time functions greatly reduces the numerical error concentrated in the corner regions, and results in very good uniformity of the ux and temperature distributions along the boundaries for problems where uniform distributions are expected.In order to highlight the importance of proper resolution both in time and space for the transient problems, we consider one-and two-dimensional formulations in this paper. High-order boundary elements using quartic shape functions, as well as high-order bi-quartic volume cells, are used to attain mesh-independent numerical solutions. We consider four transient heat di usion problems that possess exact solutions to investigate the convergence rate and accuracy of the higher-order boundary element formulations. A very high level of accuracy is possible for both one-and two-dimensional formulations. Additionally, we show that the accuracy of a commercially available ÿnite-element code is far less than that of the boundary element methods for a given spatial and temporal discretization.

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