Abstract
In this paper, we propose a BDDC preconditioner for the mortar-type rotated finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis. MSC:65N55, 65N30.
Highlights
The method of balancing domain decomposition by constraints (BDDC) was first introduced by Dohrmann in [ ]
The BDDC method is closely related to the dual-primal FETI (FETI-DP) method [ ], which is one of dual iterative substructuring methods
Each BDDC and FETI-DP method is defined in terms of a set of primal continuity
Summary
The method of balancing domain decomposition by constraints (BDDC) was first introduced by Dohrmann in [ ]. A BDDC algorithm for mortar finite element was developed in [ ], the author extended the FETI-DP algorithm to elasticity problems and Stokes problems in [ , ], respectively These algorithms are based on locally conforming finite element methods, and the coarse space components of the algorithms are related to the cross-points (i.e., corners), which are often noteworthy points in domain decomposition methods (DDMs). We study the BDDC algorithm for the mortar-type rotated Q element for the second order elliptic problem with discontinuous coefficients, where the discontinuities lie only along the subdomain interfaces. Following the technique in [ ], we construct an auxiliary discrete space and build our BDDC algorithm on an equivalent auxiliary problem This approach overcomes the difficulty caused by the mortar condition and simplifies the implementation of the BDDC preconditioning iteration. The symbols , and are used, and x y , x y , and x y mean that x ≤ C y , x ≥ C y , and c x ≤ y ≤ C y for some constants C , C , C , and c that are independent of discontinuous coefficients and mesh size
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