A THREE-DIMENSIONAL DOMAIN DECOMPOSITION ALGORITHM FOR THE NUMERICAL SOLUTION OF ELLIPTIC AND PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

A three-dimensional, nonsymmetric, domain decomposition algorithm is developed. The algorithm is based upon the use of a lower dimensional problem as a correction to the preconditioned generalized conjugate residual method using the domain decomposition technique as the overall preconditioner. For the finite difference solution of elliptic and parabolic partial differential equations with symmetric and nonsymmetric rough coefficients, the method is both robust and efficient. Three problems, including a highly heterogeneous example, an example from the SPE/SIAM comparative solution project, and a nonsymmetric parabolic reservoir simulation example, are presented to validate the method. Several comparisons are made with other well-known preconditioners including incomplete LU and reduced system/incomplete LU factorizations. For the examples considered the domain decomposition technique was the most efficient in a nonparallel environment; in a parallel computational environment the algorithm was a factor of four faster than the other techniques. An analysis is made concerning both vector and parallel computational aspects of the domain decomposition of domains, subproblem tolerances, and subproblem preconditioners on the convergence rate and computational work for the algorithm. The convergence rate is shown to be only slightly dependent on the number of subdomains. The effect of subproblem tolerances on the method is also small. Reduced system ILU(0) had the best computational efficiency for use in subproblem solutions. Corrections using the lower dimensional problem, known as line corrections, is shown to be necessary for the rapid convergence of the method. Finally, the three dimensional domain decomposition algorithm was efficiently implemented in parallel computational environments using multitasking and microtasking on both the CRAY X/MP 48 and IBM 3090/400 parallel supercomputers