GS534: Solving the 1D Poisson equation using finite differences ...

Finest level
Smoothing
Restriction
Interpolation
Coarsest level
Figure 1. Geometry for 1D multigrid
Lu = f
(12)
where L is a linear operator, u is the unknown and f is the known right-hand-side vector. If we
have an approximate solution we can define the residual
r = Lu − f
(13)
If u is the correct solution the residual is zero. We can try to minimize the residual by computing
the error v to u. Since L is linear the error satisfies
Lv =−r
(14)
so that when we know the residual we can solve (14) to find the correction. The main trick in
multigrid is that (14) is computed on a coarser grid and the coarse grid solution is interpolated to
the finer grid before adding it to the approximate fine grid solution. The two grid algorithm can
be written as
compute approximate solution on fine grid using (12)
compute residual on fine grid using (13)
restrict residual to coarse grid
solve for error on coarse grid using (14)
interpolate and add error to fine grid solution
Of course, if this works for going from one level to a next-coarser one, it will work for the next
coarser level also. This means that we can iteratively apply the coarse grid error correction. Note
that (14) and (12) have identical form so that we can define the right hand side f of each coarse
grid equation to be equal to the negative residual −r and define the solution at each coarser level u
to be the error v for the next higher level.
Assume that we want to solve (12) on a grid with 2 N + 1 grid points. This is the finest level
l = N. We can define coarse levels l = N − 1, N − 2, ..., 1 by taking out each even point from the
next finer level (figure 1). At the coarsest level we hav e three grid points left. At each level we
define a solution u l , a right hand side f l , a residual r l and error v l .
For each level l = N, N − 1, ...1
relax (12) at level l to find u l
compute negative residual on level l: −r l
interpolate residual to next coarser level and store in f l−1
4