Comparisons of Unstructured Multigrid as a Non-linear Solver, a Linear Solver, or a Pre-Conditioner

Dimitri Mavriplis
ICASE MS 132C
NASA Langley Research Center
Hampton, VA 23681


Abstract

The relative efficiency of an unstructured multigrid algorithm used as a non-linear solver (FAS) is compared with the efficiency of the equivalent multigrid algorithm used as a linear solver on a Newton linearization of the non-linear problem, and with the efficiency obtained using either approach as a preconditioner for a non-linear GMRES method. Two types of problems are examined, the solution of a transient radiation diffusion problem, and the solution of the Navier-Stokes equations. In the first case, the discretization employs a nearest neighbor stencil and the linearized scheme operates on an exact Newton linearization. In the second case, the discretization is based on a second-order form which involves neighbors of neighbors while the linearization employed is based on a first-order accurate nearest-neighbor stencil. For the radiation diffusion case, the linear multigrid approach is more efficient than the FAS approach, mainly due to the expense involved in the more frequent computation of the highly non-linear residuals required in the non-linear multigrid case. When the linear system is solved to sufficient tolerance, quadratic convergence of the non-linear problem is observed. For the Navier-Stokes equations, quadratic convergence of the non-linear problem is not possible, since an inexact linearization is employed. While a linear multigrid iteration is shown to be substantially cheaper than the corresponding non-linear multigrid iteration, the relative performance of both methods depends largely on the tolerance to which the linear system is solved, and the achievable non-linear convergence rate. Using either method as a preconditioner rather than as a solver is also shown to provide additional convergence acceleration. Other issues such as memory requirements and robustness of the various schemes will also be discussed.