An Assessment of Linear Versus Nonlinear Multigrid Methods for Unstructured Mesh Solvers

The relative performance of a nonlinear full approximation storage multigrid algorithm and an equivalent linear multigrid algorithm for solving two different nonlinear problems is investigated. The first case consists of a transient radiation diffusion problem for which an exact linearization is available, while the second problem involves the solution of the steady-state Navier–Stokes equations, where a first-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 nonlinear multigrid methods asymptotically converge at identical rates and the linear method is found to be more efficient due to its lower cost per cycle. When an approximate linearization is employed, as in the Navier–Stokes cases, the relative efficiency of the linear approach versus the nonlinear approach depends both on the degree to which the linear system approximates the full Jacobian as well as on the relative cost of linear versus nonlinear 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.