AbstractVarious advanced two-level iterative methods are studied numerically
and compared with each other in conjunction with finite volume discretizations of symmetric
1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of
the methods considered rely on the homogenization approach for deriving the coarse
grid operator. This approach is considered here as an alternative to the well-known
Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators
are studied, primary consideration being given to the use of the so-called problemdependent
prolongation. The two-grid methods considered are used as both solvers
and preconditioners for the Conjugate Gradient method. The recent approaches, such
as the hybrid domain decomposition method introduced by Vassilevski and the globallocal
iterative procedure proposed by Durlofsky et al. are also discussed. A two-level
method converging in one iteration in the case where the right-hand side is only a
function of the coarse variable is introduced and discussed. Such a fast convergence for
problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved
by a problem-dependent selection of the coarse grid combined with problem-dependent
prolongation on a dual grid. The results of the numerical experiments are presented
to illustrate the performance of the studied approaches.
Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids