The Influence of Interface Conditions on Convergence of Krylov-Schwarz Domain Decomposition for the Advection-Diffusion Equation

Several variants of Schwarz domain decomposition, which differ in the choice of interface conditions, are studied in a finite volume context. Krylov subspace acceleration, GMRES in this paper, is used to accelerate convergence. Using a detailed investigation of the systems involved, we can minimize the memory requirements of GMRES acceleration. It is shown how Krylov subspace acceleration can be easily built on top of an already implemented Schwarz domain decomposition iteration, which makes Krylov-Schwarz algorithms easy to use in practice. The convergence rate is investigated both theoretically and experimentally. It is observed that the Krylov subspace accelerated algorithm is quite insensitive to the type of interface conditions employed.

[1]  P. Tallec.  Domain decomposition methods in computational mechanics , 1994 .

[2]  Erik Brakkee,et al.  A parallel domain decomposition algorithm for the incompressible Navier-Stokes equations , 1995, Simul. Pract. Theory.

[3]  O. Widlund,et al.  Some recent results on Schwarz type domain decomposition algorithms , 1992 .

[4]  Tony F. Chan,et al.  On the Relationship between Overlapping and Nonoverlapping Domain Decomposition Methods , 1992, SIAM J. Matrix Anal. Appl..

[5]  Alfio Quarteroni,et al.  An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods , 1988 .

[6]  H. V. D. Vorst,et al.  The superlinear convergence behaviour of GMRES , 1993 .

[7]  John C. Strikwerda,et al.  A Domain Decomposition Method for Incompressible Viscous Flow , 1993, SIAM J. Sci. Comput..

[8]  Alfio Quarteroni,et al.  A relaxation procedure for domain decomposition methods using finite elements , 1989 .

[9]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[10]  Wei-Pai Tang,et al.  Generalized Schwarz Splittings , 1992, SIAM J. Sci. Comput..

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

[12]  D. W.,et al.  Multigrid on Composite Meshes , 1987 .

[13]  Rolf Rannacher,et al.  Numerical methods for the Navier-Stokes equations , 1994 .

[14]  Morten D. Skogen,et al.  Domain Decomposition Algorithms Of Schwarz Type, Designed For Massively Parallel Computers , 1992 .

[15]  Cornelis Vuik,et al.  Computing Incompressible Flows in General Domains , 1994 .

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

[17]  Olof B. Widlund,et al.  To overlap or not to overlap: a note on a domain decomposition method for elliptic problems , 1989 .

[18]  P. Wesseling,et al.  Benchmark solutions for the incompressible Navier–Stokes equations in general co‐ordinates on staggered grids , 1993 .

[19]  O. Widlund.  Some Schwarz Methods for Symmetric and Nonsymmetric Elliptic Problems , 1991 .

[20]  David E. Keyes,et al.  Aerodynamic applications of Newton- Krylov-Schwarz solvers , 1995 .

[21]  Wei Shyy,et al.  A pressure-based composite grid method for the Navier-Stokes equations , 1993 .

[22]  William Gropp,et al.  A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems , 1994, Numer. Linear Algebra Appl..