An Assessment of Linear Versus Non-linear Multigrid Methods for Unstructured Mesh Solvers

The relative performance of a non-linear FAS multigrid algorithm and an equivalent linear multigrid algorithm for solving two di erent non-linear problems is investigated. The rst case consists of a transient radiation-di usion problem for which an exact linearization is available, while the second problem involves the solution of the steady-state Navier-Stokes equations, where a rst-order discrete Jacobian is employed as an approximation to the Jacobian of a second-order accurate discretization. When an exact linearization is employed, the linear and non-linear multigrid methods converge at identical rates, asymptotically, and the linear method is found to be more e cient due to its lower cost per cycle. When an approximate linearization is employed, as in the Navier-Stokes cases, the relative e ciency of the linear approach versus the non-linear approach depends both on the degree to which the linear system approximates the full Jacobian as well as the relative cost of linear versus non-linear multigrid cycles. For cases where convergence is limited by a poor Jacobian approximation, substantial speedup can be obtained using either multigrid method as a preconditioner to a Newton-Krylov method.