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Generalizing the approach of some previous works the authors present
multilevel preconditioners for three-dimensional (3D) elliptic problems
discretized by a family of Rannacher Turek non-conforming finite
elements.
The first major contribution of the study is the derived estimates of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver, also for the case of non-smooth coefficients. The second important achievement concerns the experimental study of AMLI solvers applied to the case of voxel FEM simulation. Here the coefficient jumps are resolved on the finest mesh only and therefore the classical CBS inequality based convergence theory is not directly applicable. The obtained results, however, demonstrate the efficiency of the proposed algorithms in this case also, as it is illustrated by an example of microstructure analysis of bones. |