Convergence analysis of a balalncing domain decomposition method for solving interior Helmholtz equations

CONVERGENCE ANALYSIS OF A BALANCING DOMAIN DECOMPOSITION METHOD FOR SOLVING INTERIOR HELMHOLTZ EQUATIONS JING LI ∗ AND XUEMIN TU † Abstract. A variant of balancing domain decomposition method by constraints (BDDC) is pro- posed for solving a class of indeﬁnite system of linear equations, which arises from the ﬁnite element discretization of the Helmholtz equation of time-harmonic wave propagation in a bounded interior domain. The proposed BDDC algorithm is closely related to the dual-primal ﬁnite element tearing and interconnecting algorithm for solving Helmholtz equations (FETI-DPH). Under the condition that the diameters of the subdomains are small enough, the rate of convergence is established which depends polylogarithmically on the dimension of the individual subdomain problems and which im- proves with the decrease of the subdomain diameters. These results are supported by numerical experiments of solving a Helmholtz equation on a two-dimensional square domain. Key words. domain decomposition, preconditioner, FETI, BDDC, indeﬁnite, non-conforming, Helmholtz AMS subject classiﬁcations. 65F10, 65N30, 65N55 1. Introduction. Domain decomposition methods have been widely used and studied for solving large symmetric, positive deﬁnite linear systems arising from the ﬁnite element discretization of elliptic partial diﬀerential equations; theories on their convergence rates are well developed for such problems; see [45, 43, 41] and the ref- erences therein. Domain decomposition methods have also been applied to solving indeﬁnite and nonsymmetric problems; cf. [1, 3, 4, 7, 8, 9, 23, 28, 29, 30, 34, 35, 39, 40, 44, 46]. A two-level overlapping Schwarz method was studied by Cai and Widlund [8] for solving indeﬁnite elliptic problems, where they used a perturbation approach in the analysis to overcome the diﬃculty introduced by the indeﬁniteness of the prob- lem and established that the convergence rate of the algorithm is independent of the mesh size if the coarse level mesh is ﬁne enough. Such an approach was also used by Gopalakrishnan and Pasciak [23] and by Gopalakrishnan, Pasciak, and Demkow- icz [24] in their analysis of overlapping Schwarz methods and multigrid methods for solving time harmonic Maxwell equations. The balancing domain decomposition methods by constraints (BDDC) were in- troduced by Dohrmann [12] for solving symmetric positive deﬁnite problems; see also Fragakis and Papadrakakis [22], and Cros [11]. They represent an interesting redesign of the Neumann-Neumann algorithms with the coarse, global component expressed in terms of a set of primal constraints. Spectral equivalence between the BDDC al- gorithms and the dual-primal ﬁnite element tearing and interconnecting algorithms (FETI-DP) has been proven by Mandel, Dohrmann, and Tezaur [38]; see also Li and Widlund [36], Brenner and Sung [6]. In these papers, it is established for the sym- metric positive deﬁnite case that the preconditioned operators of a pair of BDDC and ∗ Department of Mathematical Sciences, Kent State University, Kent, OH 44242 (li@math.kent.edu, http://www.math.kent.edu/∼li/). This author’s work was supported in part by National Science Foundation contract DMS-0612574. † Department of Mathematics University of California and Lawrence Berkeley National Labora- tory, Berkeley, CA 94720-3840 (xuemin@math.berkeley.edu, http://math.berkeley.edu/∼xuemin/). This author’s work was sup- ported in part by US Department of Energy contract DE-FC02-01ER25482 and by the Director, Oﬃce of Science, Advanced Scientiﬁc Computing Research, U.S. Department of Energy under Con- tract No. DE-AC02-05CH11231.