## A unified framework for adaptive BDDC

Clemens Pechstein and Clark R. Dohrmann

### Abstract

In this theoretical study, we explore how to automate the selection of weights and primal constraints in BDDC methods for general SPD problems. In particular, we address the three-dimensional case and non-diagonal weight matrices such as the deluxe scaling. We provide an overview of existing approaches, show connections between them, and present new theoretical results: A localization of the global BDDC estimate leads to a reliable condition number bound and to a local generalized eigenproblem on each glob, i.e., each subdomain face, edge, and possibly vertex. We discuss how the eigenvectors corresponding to the smallest eigenvalues can be turned into generalized primal constraints. These can be either treated as they are or (which is much simpler to implement) be enforced by (possibly stronger) classical primal constraints. We show that the second option is the better one. Furthermore, we discuss equivalent versions of the face and edge eigenproblem which match with previous works and show an optimality property of the deluxe scaling. Lastly, we give a localized algorithm which guarantees the definiteness of the matrix $\widetilde S$ underlying the BDDC preconditioner under mild assumptions on the subdomain matrices.

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### Key words

preconditioning, domain decomposition, iterative substructuring, BDDC, FETI-DP, primal constraints, adaptive coarse space, deluxe scaling, generalized eigenvalue problems, parallel sum

### AMS subject classifications

65F08, 65N30, 65N35, 65N55

### Links to the cited ETNA articles

 [15] Vol. 45 (2016), pp. 524-544 Juan G. Calvo and Olof B. Widlund: An adaptive choice of primal constraints for BDDC domain decomposition algorithms [54] Vol. 45 (2016), pp. 75-106 Axel Klawonn, Patrick Radtke, and Oliver Rheinbach: A comparison of adaptive coarse spaces for iterative substructuring in two dimensions [117] Vol. 26 (2007), pp. 146-160 Xuemin Tu: A BDDC algorithm for flow in porous media with a hybrid finite element discretization

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