Multigrid Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 276

Starting Values

As explained in §4, for linear problems we can start on the coarsest grid and work
toward the finest one. This is also possible for many nonlinear problems, but not
for all. In particular, it can happen that the nonlinear problem only has the right
number of solutions when the discretization is sufficiently fine.

For this reason, we now assume that we have a starting value which belongs
to the domain of attraction of the desired solution. However, the error may still be
much larger than the discretization error, and in fact by several orders of magnitude.

This is the usual case in practice, and we suggest proceeding as in Algorithm
NI . However, we have first to compute an appropriate right-hand side for the
problems on the coarse grids.

In the following we use the notation of Algorithm 6.2.

6.3 Algorithm NLNI ( L ,f ,u , 0 ) for improving a starting value u , 0 for the

u equation L (u ) = f at level ≥ 0 , (such that the error of the result is of the

order of the discretization error).

u f 0 , and exit the If = 0, compute the solution ˆ of the equation L 0 (v) =

procedure.
Let > 0.
Set u − 1 , 0 = ru , 0 and

L − 1 (u − 1 , 0 ) − r L (u , 0 ) ] .

f − 1 = rf + [ ( 6 . 12 )

Find an approximate solution u − 1 of the equation L − 1 (v) = f − 1 by applying

NLNI − 1 ( L − 1 ,f − 1 ,u − 1 , 0 ) .
Determine the prolongation u , 1 = p u − 1 .

Using u , 1 as a starting value, carry out one step of the iteration NLMG . Denote
the result as u , 2 . Set

u = u , 2 .

Note that equation (6.12) has the same structure as (6.11).

Since we cannot proceed without reasonable starting values, for complicated
problems, nonlinear multigrid methods are usually combined with continuation
methods (also called incremental methods).