A BDDC Preconditioner for the Rotated FEM for Elliptic Problems with Discontinuous Coefficients

We propose a BDDC preconditioner for the rotated finite element method for second order elliptic equations with piecewise but discontinuous coefficients. 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 quasioptimal convergence behavior; that is, 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. 1. Introduction The balancing domain decomposition by constraints (BDDC) method was first introduced by Dohrmann in [1]. Then Mandel and the author Dohrmann restated the method in an abstract manner and provided its convergence theory in [2]. The BDDC method is closely related to the dual-primal FETI (FETI-DP) method [3], which is one of dual iterative substructuring methods. Each BDDC and FETI-DP method is defined in terms of a set of primal continuity; the primal continuity is enforced across the interface between the subdomains and provides a coarse space component of the preconditioner. In [4], Mandel et al. analyzed the relation between the two methods and established the corresponding theory. In the last decades, the two methods have been widely analyzed and successfully been extended to many different types of partial differential equations. In [3], the two algorithms for elliptic problems were rederived and a brief proof of the main result was given. A BDDC algorithm for mortar finite element was developed in [5]; meanwhile, the authors also extended the FETI-DP algorithm to elasticity problems and stokes problems in [6, 7], respectively. These algorithms were based on locally conforming finite element methods, and the coarse space components of the algorithms were related to the cross-points (i.e., corners), which are often noteworthy points in domain decomposition methods (DDMs). Since the cross-points are related to more than two subregions, thus it is not convenient when designing algorithm. The BDDC method derives from the Neumann-Neumann domain decomposition method (see [8]). The difference is that the BDDC method applies an additive rather than a multiplicative coarse grid correction, and substructure spaces have some constraints which result in nonsingular subproblems, so that we can solve each subproblem and coarse problem in parallel. The rotated element is an important nonconforming element. It was introduced by Rannacher and Turek in [9]