PEACEMAN-RACHFORD PROCEDURE AND DOMAIN DECOMPOSITION FOR FINITE-ELEMENT PROBLEMS

This paper presents a general method to associate the operator splitti ng for the Peaceman-Rachford procedure with a decomposition of the dom ain in problems arising from finite element discretization of partial differential equations. The algorithm is provably convergent without a ny symmetry requirement. Moreover, this method possesses the significa nt advantage of making the linear systems of the Peaceman-Rachford ite ration block diagonal and therefore perfectly appropriate for parallel processing. Not only is sparsity not affected but a reduction of the bandwidth occurs. In fact, for appropriate choices of nonconforming fi nite element spaces, this method makes directly possible elementwise p rocessing. This option remains available in general for higher-dimensi onal problems by applying the splitting algorithm recursively. Practic al implementation requires nothing more than the standard finite eleme nt assembly procedure and some bookkeeping to relate a few different o rderings of the nodes. In addition to all these attractive features, t he method is rapidly convergent and remains highly competitive even wh en used on a serial machine.