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Preconditioners based on various multilevel extensions of two-level piecewise linear finite element methods lead to iterative methods which have an optimal order computational complexity with respect to the size (or discretization parameter) of the system. The methods can be in block matrix factorized form, recursively extended via certain matrix polynomial approximations of the arising Schur complement matrices or on additive, i.e., block diagonal form using stabilizations of the condition number at certain levels. The resulting spectral equivalence holds uniformly with respect to jumps in the coefficients of the differential operator and for arbitrary triangulations. Such methods were first presented by Axelsson and Vassilevski in the late 1980s. An important part of the algorithm is the treatment of systems with a diagonal block matrix, which arises on each finer level in a recursive refinement method and corresponds to the added degrees of freedom on that level. This block is well-conditioned for model type problems but becomes increasingly ill-conditioned when the coefficient matrix becomes more anisotropic or, equivalently, when the mesh aspect ratio increases. This paper presents some methods for approximating this matrix also leading to a preconditioner with spectral equivalence bounds which hold uniformly with respect to both the problem and the discretization parameters. Therefore, the same holds also for the preconditioner to the global matrix.

On Multilevel Preconditioners which are Optimal with Respect to Both Problem and Discretization Parameters