[petsc-users] Multigrid preconditioner parameters

[Jed Brown]

Feng Xing <snakexf at gmail.com> writes:

> In detail, the equation is div( K(x) grad(u) ) = f, where K(x) could
> be a variable coefficient or a constant, in a nice domain
> ((0,1)^3). The problem is defined on a nice domain like
> (0,1)^3. Laplace is a special case if K(x)=cst. So I wanted to try to
> understand it (convergence of multigrid ).

Be aware that the coefficient structure fundamentally changes the
behavior of the equation and the expected convergence properties.  With
coefficient jumps of 10^10 and fine structure with long correlation
length, the discretization will always be under-resolved.  Depending on
the variation and geometry, it can be easy to solve or very difficult.

One way to understand this: think of a large maze drawn in the plane.
Set Dirichlet u=0 and u=1 boundary conditions at the start and end, with
Neumann on all other surfaces, K(x) = \epsilon on walls, and K(x)=1
everywhere else.  Solve this elliptic problem -- the solution of the
maze just follows the gradient (all corridors leading to dead-ends are
constant).  Obviously this is highly non-local -- small far-away changes
in K(x) completely changes the solution.  Nothing like what you're used
to if you solve smooth Laplacians.
-------------- next part --------------
A non-text attachment was scrubbed...
Name: signature.asc
Type: application/pgp-signature
Size: 818 bytes
Desc: not available
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20150529/36b9d9d9/attachment.pgp>