SIAM J. NUMER. ANAL. c  2006 Society for Industrial and Applied Mathematics Vol. 44, No. 1, pp. 283–299 A FETI-DP PRECONDITIONER WITH A SPECIAL SCALING FOR MORTAR DISCRETIZATION OF ELLIPTIC PROBLEMS WITH DISCONTINUOUS COEFFICIENTS ∗ N. DOKEVA † , M. DRYJA ‡ , AND W. PROSKUROWSKI † Abstract. We consider two-dimensional elliptic problems with discontinuous coefficients dis- cretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a dual-primal FETI method. In this paper we introduce and analyze a preconditioner with a special scaling of coefficients and step parameters and establish convergence bounds. We show that the preconditioner is almost optimal with constants independent of the jumps of coefficients and step parameters. Extensive computational evidence is presented that illustrates an almost optimal convergence for a variety of situations (distribution of subregions, grid assignment, grid ratios, number of subregions) for both continuous and discontinuous problems. Key words. domain decomposition, mortar finite element method, dual-primal FETI precon- ditioner, nonmatching grids, saddle-point problem, elliptic problems with discontinuous coefficients AMS subject classifications. 65N55, 65N30, 65F10 DOI. 10.1137/040616401 1. Introduction. In this paper we discuss a second order elliptic problem with discontinuous coefficients defined on a polygonal region Ω ⊂ R 2 which is a union of many polygons Ω i . The problem is discretized by the finite element method (FEM) on geometrically conforming nonmatching triangulations across Γ= ∪ i ∂ Ω i \∂ Ω using the mortar technique; see [1]. The resulting discrete problem is solved by a dual-primal FETI (FETI-DP) method; see [5], [6], [7] for the matching triangulation and [3], [4] for the nonmatching one. The method is discussed under the assumption of continuity of the solution at vertices of Ω i . We prove that the method is convergent and its rate of convergence is almost optimal and independent of the jumps of coefficients, provided that a mortar side is associated with the higher coefficient. Consequently, the method is well suited for parallel processors. The presented results are a generalization of results obtained in [4] and [3] for problems with continuous and discontinuous coefficients, respectively. In [4] a modi- fied mortar condition at the vertices of substructures is employed using the assumption that the solution at the vertices is continuous, while in [3] a standard approximation to the mortar condition is employed. The preconditioner in [3] which does not use the scaling of the coefficients was tested for the simplest case of four subregions. In general, however, the experiments show that for discontinuous coefficients the precon- ditioner without proper scaling of coefficients exhibits poor convergence. ∗ Received by the editors October 5, 2004; accepted for publication (in revised form) June 28, 2005; published electronically March 7, 2006. http://www.siam.org/journals/sinum/44-1/61640.html † Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113 (dokeva@usc.edu, proskuro@math.usc.edu). ‡ Department of Mathematics, Warsaw University, Banach 2, 02-097 Warsaw, Poland (dryja@ mimuw.edu.pl). The work of this author was supported in part by the U.S. Department of Energy under contract DE-FG02-92ER25127 and in part by the Polish Science Foundation under grant 2P03A00524. 283