A FETI method with a mesh independent condition number for the iteration matrix

Abstract

We introduce a framework for FETI methods using ideas from the decomposition via Lagrange multipliers of H 0 1 ( Ω ) derived by Raviart–Thomas [P.A. Raviart, J.-M. Thomas, Primal hybrid finite element methods for second order elliptic equations, Math. Comput. 31 (138) (1977) 391–413] and complemented with the detailed work on polygonal domains developed by Grisvard [P. Grisvard, Singularities in boundary value problems, Rech. Math. Appl. 22 (1992)]. We compute the action of the Lagrange multipliers using the natural H 00 1 / 2 scalar product, therefore no consistency error appears. Our analysis allows to deal with cross points and floating subdomains in a natural manner. As a byproduct, we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning. This result improves the standard asymptotic bound for this condition number given by ( 1 + log ( H / h ) ) 2 , where H and h are the characteristic subdomain size and element size respectively, shown by Mandel–Tezaur in [J. Mandel, R. Tezaur, Convergence of a substructuring method with Lagrange multipliers, Numer. Math. 73 (1996) 473–487]. Numerical results that confirm our theoretical analysis are presented.