A Domain Decomposition Solver for the Steady Navier-Stokes Equations

We present two new domain decomposition solvers in the context of conforming spectral element discretizations. The first is a domain decomposition solver for the discrete steady convection-diffusion equation, while the second is a domain decomposition solver for the discrete steady Stokes or Navier-Stokes equations. The solution algorithms are both based on the additive Schwarz method in the context of nonoverlapping subdomains. The key ingredients are: (i) a coarse global system; (ii) a set of local, independent subproblems associated with the subdomains (or spectral elements): (iii) a system associated with the unknowns on the subdomain interfaces; and (iv) a Krylov method such as the CG algorithm or the GMRES algorithm. We present numerical results that demonstrate the convergence properties of the new solvers, as well as the applicability of the methods to solve heat transfer and incompressible fluid flow problems.

[1]  Michel O. Deville,et al.  Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems , 1990, SIAM J. Sci. Comput..

[2]  Olof B. Widlund,et al.  A Polylogarithmic Bound for an Iterative Substructuring Method for Spectral Elements in Three Dimensions , 1996 .

[3]  Anthony T. Patera,et al.  Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations , 1993, SIAM J. Sci. Comput..

[4]  Marek Behr,et al.  Finite element solution strategies for large-scale flow simulations☆ , 1994 .

[5]  Barry F. Smith,et al.  Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions , 1994 .

[6]  Olof B. Widlund,et al.  Domain Decomposition Algorithms with Small Overlap , 1992, SIAM J. Sci. Comput..

[7]  M. Deville,et al.  Chebyshev pseudospectral solution of the Stokes equations using finite element preconditioning , 1989 .

[8]  Paul Fischer,et al.  Spectral element methods for large scale parallel Navier—Stokes calculations , 1994 .

[9]  A. Patera.  A spectral element method for fluid dynamics: Laminar flow in a channel expansion , 1984 .

[10]  Tony F. Chan,et al.  Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes , 1994 .

[11]  L. Kovasznay.  Laminar flow behind a two-dimensional grid , 1948 .

[12]  Michel Fortin,et al.  EXPERIMENTS WITH SEVERAL ELEMENTS FOR VISCOUS INCOMPRESSIBLE FLOWS , 1985 .

[13]  P. Fischer,et al.  Parallel spectral element solution of the Stokes problem , 1991 .

[14]  Steven A. Orszag,et al.  Spectral Methods for Problems in Complex Geometrics , 1979 .

[15]  Barry Smith.  A Parallel Implementation of an Iterative Substructuring Algorithm for Problems in Three Dimensions , 1993, SIAM J. Sci. Comput..

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

[17]  J. Reddy,et al.  Accuracy and convergence of element-by-element iterative solvers for incompressible fluid flows using penalty finite element model , 1993 .

[18]  Olof B. Widlund,et al.  Iterative Substructuring Methods for Spectral Elements: Problems in Three Dimensions Based on Numerical Quadrature , 1997 .

[19]  Tayfun E. Tezduyar,et al.  Clustered element-by-element computations for fluid flow , 1992 .