This paper considers several moving mesh partial differential equations that are related to the equidistribution principle. Several of these are new, and some correspond to discrete moving mesh equations that have been used by others. Their stability is analyzed and it is seen that a key term for most of these moving mesh PDEs is a source-like term that measures the level of equidistribution. It is shown that under weak assumptions mesh crossing cannot occur for most of them. Finally, numerical experiments for these various moving mesh PDEs are performed to study their relative properties.

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Moving mesh methods based on the equidistribution principle are studied from the viewpoint of stability of the moving mesh system of di erential equations. For ne spatial grids, the moving mesh system inherits the stability of the original discretized PDE. Unfortunately, for some PDEs the moving mesh methods require so many spatial grid points that they no longer appear to be practical. Failures and successes of the moving mesh method applied to three reaction-di usion problems are explained via an analysis of the stability and accuracy of the moving mesh PDE.

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Abstract. Dense linear algebra libraries need to cope efficiently with a range of input problem sizes and shapes. Inherently this means that parallel implementations have to exploit parallelism wherever it is present. While OpenMP allows relatively fine grain parallelism to be exploited in a shared memory environment it currently lacks features to make it easy to partition computation over multiple array indices or to overlap sequential and parallel computations. The inherent flexible nature of shared memory paradigms such as OpenMP poses other difficulties when it becomes necessary to optimise performance across successive parallel library calls. Notions borrowed from distributed memory paradigms, such as explicit data distributions help address some of these problems, but the focus on data rather than work distribution appears misplaced in an SMP context.