Computational Methods in Applied Mathematics Vol. 12 (2012), No. 1, pp. 73–91 c ⃝ 2012 Institute of Mathematics, NAS of Belarus Doi: 10.2478/cmam-2012-0004 A FETI-DP Method for Crouzeix-Raviart Finite Element Discretizations Leszek Marcinkowski · Talal Rahman Abstract — This paper is concerned with the construction and analysis of a paral- lel preconditioner for a FETI-DP system of equations arising from the nonconforming Crouzeix-Raviart finite element discretization of a model elliptic problem of second order with discontinuous coefficients. We show that the condition number of the preconditioned problem is independent of the coefficient jumps, and grows only as (1 + log(H/h)) 2 , where H and h are mesh parameters, in other words the precondi- tioner is quasi optimal. 2010 Mathematical subject classification: 65N55; 65N30; 65F08. Keywords: FETI-DP method; Crouzeix-Raviart nonconforming finite element method; domain decomposition; elliptic differential equations of second order. 1. Introduction In many scientific applications, where partial differential equations are used to model, the Crouzeix-Raviart (CR) finite element [18] appears to be among the most commonly used non- conforming finite element for the discretization, which includes applications like the Poisson problem, the Stokes or the Navier-Stokes problem (cf. [18, 38]), the Darcy-Stokes problem (cf. [15], the elasticity problem (cf. [5, 22]), and problems on nonmatching grids (cf. [30, 34]). There is also a close relationship between mixed finite elements and the nonconforming finite element for the second order elliptic problem which makes the CR finite element interesting; cf. [1, 2]. The element has also been used in the framework of finite volume element method; cf. [16]. There are a number of effective solvers of the domain decomposition type for the CR finite element which can be found in the literature, see, e.g., [7, 17, 26, 31, 36, 37] for work on two level algorithms, [23, 33, 35] on multilevel algorithms, [4, 6] on multigrid algorithms, and [32] on substructuring type algorithms. To our knowledge, however, there is not any work on FETI-DP type domain decomposition algorithms for the Crouzeix-Raviart (CR) finite element. The FETI-DP methods form a class of fast and efficient iterative solvers for the algebraic systems arising from the finite element discretization of partial differential equations of sec- ond and fourth order; see, e.g., [21, 25, 28, 29, 19, 20, 24, 10, 11, 14] and references therein. In this paper, we introduce a FETI-DP method for solving the system of equations arising Leszek Marcinkowski Faculty of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland E-mail: L.Marcinkowski@mimuw.edu.pl. Talal Rahman Department of Computer Engineering, Bergen University College, Nyg˚ ardsgaten 112, N-5020 Bergen, Nor- way E-mail: Talal.Rahman@hib.no Brought to you by | University of Calgary Authenticated Download Date | 5/25/15 9:21 AM