SUBSTRUCTURING DOMAIN DECOMPOSITION METHOD FOR NONCONFORMING FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC PROBLEMS WITH ANISOTROPY

Abstract. An optimal iterative method for solving systems of linear algebraic equations arising from nonconforming nite element discretizations of second order elliptic boundary value problems with anisotropic coe cients is constructed and studied. The technique suggested is based on decomposition of the original domain into nonoverlapping subdomains. It is assumed that the equation coe cients in each subdomain vary inconsiderably. This approach combines the ideas of domain decomposition methods, the algorithms of multilevel and algebraic multigrid methods with the bordering method. An iterative process whose convergence rate is independent of the mesh step size and the ratio of anisotropy in the coe cients is constructed. It is shown that the number of arithmetic operations required for realization of the method with a given accuracy is proportional to the number of unknowns of the original algebraic problem. Key words. Second order elliptic problem, nonconforming nite element method, domain decomposition method, algebraic substructuring preconditioner, separable matrix, condition number. AMS subject classi cations. 65N30, 65N22, 65F10. In this paper we proposeaniterative method for solving systems of grid equations approximating boundary value problems for second order elliptic equations with anisotropic coe cients. The technique suggested is based on decomposition of the original domain compose