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.