A comparison of adaptive coarse spaces for iterative substructuring in two dimensions

Axel Klawonn, Patrick Radtke, and Oliver Rheinbach

Abstract

The convergence rate of iterative substructuring methods generally deteriorates when large discontinuities occur in the coefficients of the partial differential equations to be solved. In dual-primal Finite Element Tearing and Interconnecting (FETI-DP) and Balancing Domain Decomposition by Constraints (BDDC) methods, sophisticated scalings, e.g., deluxe scaling, can improve the convergence rate when large coefficient jumps occur along or across the interface. For more general cases, additional information has to be added to the coarse space. One possibility is to enhance the coarse space by local eigenvectors associated with subsets of the interface, e.g., edges. At the center of the condition number estimates for FETI-DP and BDDC methods is an estimate related to the $P_D$-operator which is defined by the product of the transpose of the scaled jump operator $B_D^T$ and the jump operator $B$ of the FETI-DP algorithm. Some enhanced algorithms immediately bring the $P_D$-operator into focus using related local eigenvalue problems, and some replace a local extension theorem and local Poincaré inequalities by appropriate local eigenvalue problems. Three different strategies, suggested by different authors, are discussed for adapting the coarse space together with suitable scalings. Proofs and numerical results comparing the methods are provided.

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Key words

FETI-DP, BDDC, eigenvalue problem, coarse space, domain decomposition, multiscale

AMS subject classifications

65F10, 65N30, 65N55