Multigrid methods nonlinear problems: an overview

Since their early application to elliptic partial differential equations, multigrid methods have been applied successfully to a large and growing class of problems, from elasticity and computational fluid dynamics to geodetics and molecular structures. Classical multigrid begins with a two-grid process. First, iterative relaxation is applied, whose effect is to smooth the error. Then a coarse-grid correction is applied, in which the smooth error is determined on a coarser grid. This error is interpolated to the fine grid and used to correct the fine-grid approximation. Applying this method recursively to solve the coarse-grid problem leads to multigrid. The coarse-grid correction works because the residual equation is linear. But this is not the case for nonlinear problems, and different strategies must be employed. In this presentation we describe how to apply multigrid to nonlinear problems. There are two basic approaches. The first is to apply a linearization scheme, such as the Newton's method, and to employ multigrid for the solution of the Jacobian system in each iteration. The second is to apply multigrid directly to the nonlinear problem by employing the so-called Full Approximation Scheme (FAS). In FAS a nonlinear iteration is applied to smooth the error. The full equation is solved on the coarse grid, after which the coarse-grid error is extracted from the solution. This correction is then interpolated and applied to the fine grid approximation. We describe these methods in detail, and present numerical experiments that indicate the efficacy of them.