Feng Xing <[email protected]> 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.
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