An Abstract Data Type for Parallel Simulations Based on Sparse Grids

In this paper we introduce an abstract data type for the distributed representation and eecient handling of sparse grids on parallel architectures. The new data layout, implemented by means of the PVM message passing library, provides partitioning and dynamic load balancing transparent to the application making use of it. This enables parallelization of sequential partial diierential equations (PDE) solvers based on sparse grids with hardly any source code modiications. 1 Sparse Grids The numerical simulation of problems occuring in engineering and natural sciences need quite ne grids for discretization if they should supply reasonable results. The concept of adaptivity takes care of smoothness of the treated function, provides a grid which is appropriate to the given problem and allows eecient discretization. Adaptive full grids require only few grid points where the solution is smooth and possess many grid points where regularity vanishes. Sparse grids were introduced 1990 by 7] (see also 2]) and represent a given function (e.g. a solution of a PDE) with even less grid points than adaptive full grids do. Moreover, sparse grids allow the concept of adaptivity, too. Sparse grids only contain O(h ?1 l (ld(h ?1 l)) d?1) grid points compared to O(h ?d l) for usual full grids, where d denotes the dimension of the problem and h l the mesh width on the boundary. On the other hand the interpolation error in the L 2 norm only increases slightly from order O(h 2 l) to O(h 2 l (ld(h ?1 l)) d?1). Sparse grid points are a subset of grid points of a full grid. An important characteristic ? This work is being funded by the ministry for education, science, research and technology BMBF within the project PAR-CVD (Parallel simulation of Chemical Vapor Deposition processes)