Title: On the Design of Small Coarse Spaces for Domain Decomposition Algorithms
Author(s): Dohrmann, Clark; Widlund, Olof
Methods are presented for automatically constructing %low-dimensional coarse spaces of low dimension for domain decomposition algorithms. These constructions use equivalence classes of nodes on the interface between the subdomains into which the domain of a given elliptic problem has been subdivided, e.g., by a mesh partitioner such as METIS; these equivalence classes already play a central role in the design, analysis, and programming of many domain decomposition algorithms. The coarse space elements are well defined even for irregular subdomains, are continuous, and well suited for use in two-level or multi-level preconditioners such as overlapping Schwarz algorithms. An analysis for scalar elliptic and linear elasticity problems ms reveals that significant reductions in the coarse space dimension can be achieved while not sacrificing the favorable condition number estimates for larger coarse spaces previously developed. These estimates depend primarily on the Lipschitz parameters of the subdomains. Numerical examples for problems in three dimensions are presented to illustrate the methods and to confirm the analysis. In some of the experiments, the coefficients have large discontinuities across the interface between the subdomains, and in some, the subdomains are generated by mesh partitioners.
Title: Isogeometric BDDC Deluxe Preconditioners for Linear Elasticity
Author(s): Pavarino, Luca F.; Scacchi, Simone; Widlund, Olof B.; Zampini, Stefano
Balancing domain decomposition by constraints (BDDC) preconditioners have been shown to provide rapidly convergent preconditioned conjugate gradient methods for solving many of the very ill-conditioned systems of algebraic equations which often arise in finite element approximations of a large variety of problems in continuum mechanics. These algorithms have also been developed successfully for problems arising in isogeometric analysis. In particular, the BDDC deluxe version has proven very successful for problems approximated by non-uniform rational B-splines (NURBS), even those of high order and regularity. One main purpose of this paper is to extend the theory, previously fully developed only for scalar elliptic problems in the plane, to problems of linear elasticity in three dimensions. Numerical experiments supporting the theory, are also reported. Some of these experiments highlight the fact that the development of the theory can help to decrease substantially the dimension of the primal space of the BDDC algorithm, which provides the necessary global component of these preconditioners, while maintaining scalability and good convergence rates.