SUMMARYThis paper summarizes the results from a special session dedicated to understanding the uid dynamics of the 8:1 thermally driven cavity which was held at the First MIT Conference on Computational Fluid and Solid Dynamics in June, 2001. The primary objectives for the special session were to: (1) determine the most accurate estimate of the critical Rayleigh number above which the ow is unsteady, (2) identify the correct, i.e. best time-dependent benchmark solution for the 8:1 di erentially heated cavity at particular values of the Rayleigh and Prandtl numbers, and (3) identify those methods that can reliably provide these results.

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SUMMARYThis paper presents a detailed multi-methods comparison of the spatial errors associated with ÿnite difference, ÿnite element and ÿnite volume semi-discretizations of the scalar advection-di usion equation. The errors are reported in terms of non-dimensional phase and group speed, discrete di usivity, artiÿcial di usivity, and grid-induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew-symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure di usion. It is demonstrated that streamline upwind Petrov-Galerkin and its control-volume ÿnite element analogue, the streamline upwind control-volume method, produce both an artiÿcial di usivity and a concomitant phase speed adjustment in addition to the usual semi-discrete artifacts observed in the phase speed, group speed and di usivity. The Galerkin ÿnite element method and its streamline upwind derivatives are shown to exhibit super-convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second-order behaviour. In Part II of this paper, we consider two-dimensional semi-discretizations of the advection-di usion equation and also assess the a ects of grid-induced anisotropy observed in the non-dimensional phase speed, and the discrete and artiÿcial di usivities. Although this work can only be considered a ÿrst step in a comprehensive multimethods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.

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SUMMARYIn an attempt to overcome some of the well-known 'problems' with the QlPo element, we have devised two 'stabilized' versions of the QIQl element, one based on a semi-implicit approximate projection method and the other based on a simple forward Euler technique. While neither one conserves mass in the most desirable manner, both generate a velocity field that is usually 'close enough' to divergence-free. After attempting to analyse the two algorithms, each of which includes some ad hoc 'enhancements', we present some numerical results to show that they both seem to work well enough. Finally, we point out that any projection method that uses a 'pressure correction' approach is inherently limited to time-accurate simulations and, even if treated fully implicitly, is inappropriate for seeking steady states via large time steps.

The research summarized in this paper is part of a multi-year effort focused on evaluating the viability of wavelet bases for the solution of partial differential equations. The primary objective for this work has been to establish a foundation for hierarchical/wavelet simulation methods based upon numerical performance, computational ef®ciency, and the ability to exploit the hierarchical adaptive nature of wavelets. This work has demonstrated that hierarchical bases can be effective for problems with a dominant elliptic character. However, the strict enforcement of orthogonality in the usual L 2 sense is less desirable than orthogonality in the energy norm. This conclusion has led to the development of a multi-scale linear ®nite element based on a hierarchical change-of-basis. This work considers the numerical and computational performance of the hierarchical Schauder basis in a Galerkin context. A unique row-column lumping procedure is developed with multi-scale solution strategies for 1-D and 2-D elliptic partial differential equations.