Elliptic boundary value problems with diffusion coefficients that rapidly and strongly vary on fine scales pose severe problems to standard multilevel methods. This is because those methods construct coarse grid operators which do not properly approximate the coarse scale structure of the problem. For instance, even if the fine scale problem is isotropic the coarse scale operator can be anisotropic.
In analogy to the asymptotic case, the construction of well-chosen coarse grid operators is called numerical homogenization. Recently, several researchers discovered that, while standard multilevel methods suffer, the multiscale structure itself is ideally suited for accomplishing that task. In principle, three approaches have been suggested:
By analyzing all these drawbacks the author has realized the importance of conserving from fine to coarse scale those properties which are continuous with respect to the so-called H-convergence of diffusion matrices. This analysis has led to a new, solution-dependent way of constructing coarse grid operators, iteratively within a multigrid cycle. Numerical examples will show the promising features of the new approach.