Automated adaptive multilevel solver

Abstract This paper presents an automated adaptive multilevel solver for linear (or linearized) system of equations. The multilevel aspect of the solver is aimed at securing an optimal rate of convergence, while keeping the size of the coarsest problem sufficiently small to ensure that the direct portion of the solution does not dominate the total computational cost. Adaptivity in terms of a priori selection of the number of levels (one or more) and construction of the optimal multilevel preconditioner is the key to the robustness of the method. The number of levels is selected on the basis of estimated conditioning, sparsity of the factor and available memory. The auxiliary coarse models (if required) are automatically constructed on the basis of spectral characteristics of individual aggregates (groups of neighboring elements). An obstacle test consisting of twenty industry and model problems was designed to (i) determine the optimal values of computational parameters and to (ii) compare the adaptive multilevel scheme with existing state-of-the-art equation solvers including the Multifrontal solver [17] with the MMD reordering scheme, and the PCG solver with the nearly optimal Modified Incomplete Cholesky factorization preconditioner.

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