Scalable algorithms for the solution of Navierʹs equations of elasticity

Very large scale mathematical modelling such as accurate modelling using Navierʹs equations of elasticity needs both massively parallel computing and scalable algorithms. It is shown in this paper that efficient methods must be scalable with respect to the speedup measured as the ratio of the computing time of the best sequential algorithm on one processor and the computing time of the parallel algorithm on p processors. For a class of multilevel methods for elliptic partial differential equations it is shown how to balance the coarsest mesh size to the finest and the number of processors to the size of the problem to get smallest computing time and maximal efficiency. It turns out that the number of processors should grow slowly in proportion with the problem size. Further, it should grow slightly slower (by a logarithmic or a polylogarithmic function) for asymptotically maximal efficiency than the number of processors required for minimal computing time.