Domain Decomposition Techniques for Large Sparse Nonsymmetric Systems Arising from Elliptic Problems with First-Order Terms

Parallel block-preconditioned domain-decomposed Krylov methods for sparse linear systems are described and illustrated on two-dimensional model problems of algebraic dimension up to 65,025. Four convective-diffusive transport problems typical of implicit upwind finite-difference discretizations of heat and mass transfer applications (pure conduction, a plug flow, a jet flow, and a recirculating flow) are tested for practicality of parallel solution under the domain decomposition paradigm. The discrete operators corresponding to the latter two lack constant coefficients and symmetry, and there is little iterative convergence theory to guide their solution, but much practical progress can be made. We describe techniques depending only on the sparsity structure and approximate diagonal dominance of the linear operator and thus of broad applicability. Results of tests run on an Encore Multimax with up to 16 processors demonstrate their utility in the coarse-granularity parallelization of hydrocodes.

[1]  H. Walker.  Implementation of the GMRES method using householder transformations , 1988 .

[2]  David E. Keyes,et al.  Domain decomposition methods for the parallel computation of reacting flows , 1989 .

[3]  William Gropp,et al.  A comparison of domain decomposition techniques for elliptic partial differential equations and their parallel implementation , 1985, PP.

[4]  T. Chan,et al.  A framework for the analysis and construction of domain decomposition preconditioners , 1988 .

[5]  R. Cottle.  Manifestations of the Schur complement , 1974 .

[6]  R. Glowinski,et al.  Computing Methods in Applied Sciences and Engineering , 1974 .

[7]  Olof B. Widlund.  Some Domain Decomposition and Iterative Refinement Algorithms for Elliptic Finite Element Problems , 1988 .

[8]  M. Powell,et al.  On the Estimation of Sparse Jacobian Matrices , 1974 .

[9]  T. Manteuffel.  The Tchebychev iteration for nonsymmetric linear systems , 1977 .

[10]  H. Elman,et al.  Ordering techniques for the preconditioned conjugate gradient method on parallel computers , 1989 .

[11]  J. S. Przemieniecki.  Matrix Structural Analysis of Substructures , 1963 .

[12]  R. Glowinski,et al.  Third International Symposium on Domain Decomposition Methods for Partial Differential Equations , 1990 .

[13]  J. Meijerink,et al.  Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems , 1981 .

[14]  Y. Saad,et al.  Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems , 1984 .

[15]  H. Schlichting.  Boundary Layer Theory , 1955 .

[16]  W. James,et al.  A Conjugate Gradient-Truncated Direct Method for the Iterative Solution of the Reservoir Simulation Pressure Equation , 1981 .

[17]  William Gropp,et al.  Complexity of Parallel Implementation of Domain Decomposition Techniques for Elliptic Partial Differential Equations , 1988 .

[18]  O. Axelsson,et al.  On the eigenvalue distribution of a class of preconditioning methods , 1986 .

[19]  R. P. Kendall,et al.  An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations , 1968 .

[20]  Youcef Saad,et al.  Parallel Implementations of Preconditioned Conjugate Gradient Methods. , 1985 .

[21]  Tony F. Chan,et al.  A domain-decomposed fast Poisson solver on a rectangle , 1985, PPSC.

[22]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[23]  W. Proskurowski,et al.  Capacitance matrix method using strips with alternating neumann and dirichlet boundary conditions , 1985 .

[24]  T. Chan,et al.  A Survey of Preconditioners for Domain Decomposition. , 1985 .

[25]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .