Solving 3D Incompressible Navier-Stokes via Parallel Block Preconditioning

Victoria E. Howle

Sandia National Laboratories
PO Box 969, MS 9217
Livermore, CA 94550

Ray Tuminaro

Sandia National Laboratories
PO Box 969, MS 9217
Livermore, CA 94550

John Shadid

Sandia National Laboratories
PO Box 5800, MS 1111
Albuquerque, NM 87185

Howard Elman

Department of Computer Science
University of Maryland
College Park, MD 20742


Abstract

We consider parallel steady state and time dependent solutions of the 3D Lid Driven Cavity problem. This talk focuses mainly on two issues in resolving the set of nonlinear equations that result from discretization: (1) nonlinear solver performance and how different nonlinear solvers yield vastly different linear subproblem characteristics; (2) solution of the linear subproblems via block preconditioning along the lines of Kay & Loghin, Elman, and Silvester & Wathen. Specifically, both Newton's method and a fixed point technique based on an Oseen iteration are considered for the nonlinear problem. Block preconditioning methods using approximate Schur complement operators in conjuction with algebraic multigrid for subproblems will be discussed to solve the Oseen submatrix problems. For Newton's method, two preconditioners are considered: one applied to an explicitly calculated Jacobian matrix and the other applied to the Oseen matrix. We illustrate good convergence rates, similar to those observed by other authors, for two dimensional steady state and time dependent problems, time dependent three dimensional problems, and low Reynolds number steady state three dimensional problems. However, the steady state large Reynolds number 3D lid driven cavity problem is significantly more problematic. We will illustrate the utility of these ideas with a variety of benchmark problems and discuss some of the difficulties associated solution of steady 3D problems with large Reynolds numbers, as well as possible remedies.