A robust and parallel relaxation method based on algebraic splittings

We propose and analyze a new relaxation scheme for the iterative solution o f the linear system arising from the finite difference discretization of co nvection-diffusion problems. For problems that are convection dominated, th e (nondimensionalized) diffusion parameter epsilon is usually several order s of magnitude smaller than computationally feasible mesh widths. Thus, it is of practical importance that approximation methods not degrade for small epsilon. We give a relaxation procedure that is proven to converge uniform ly in epsilon to the solution of the linear algebraic system (i.e., "robust ly"). The procedure requires, at each step, the solution of one 4 x 4 linea r system per mesh cell. Each 4 x 4 system can be independently solved, and the result communicated to the neighboring mesh cells. Thus, on a mesh conn ected processor array, the communication requirements are four local commun ications per iteration per mesh cell. An example is given, which illustrate s the robustness of the new relaxation scheme. (C) 1999 John Wiley & Sons, Inc.