: The macro--hybrid formulation based on domain decomposition is considered for elliptic boundary value problems with both symmetric positive definite and indefinite operators. The problem is discretized by the mortar element method, which leads to a large--scale sparse linear system with a saddle-- point matrix. In the case of symmetric and positive definite operators, a block diagonal preconditioner based on fictitious domains is proposed, which is spectrally equivalent to the original saddle--point matrix. In the case of the Helmholtz operator with the absorbing boundary conditions, a special preconditioner is introduced such that the subspace of constraints remains invariant with respect to the preconditioned GMRES method. Results of numerical experiments are presented. Keywords: Macro--hybrid formulation, domain decomposition, saddle--point problem, Lagrange multipliers, fictitious domain method, nonmatching grids. Subject classification (AMS): 65F10, 65N22 1 Introduction In th...

. Qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal Stieltjes-Toeplitz matrices are studied. Two particular splittings, the so-called symmetric tridiagonal splittings and the bidiagonal splittings, are considered, and conditions for qualitative properties like nonnegativity and shape preservation are shown for them. A special attention is paid to their close relation to the well-known splitting techniques like regular and weak regular splitting methods. Extensions to block tridiagonal matrices are given as well as their relation to algebraic representations of domain decomposition methods are discussed. The paper is concluded with illustrative numerical experiments. Keywords. Matrix splitting methods, Stieltjes-Toeplitz matrices, qualitative analysis, SOR method, regular and weak regular splittings, domain decomposition AMS Subject Classification. 65F10, 65M60, 15A48 1 Introduction In this paper, we study matrix splitting me...

. Two-stage algebraic models with symmetric tridiagonal Toeplitz matrices and their qualitative properties are studied. Their relation to the well-known one-step iteration processes as well as the matrix splitting methods are pointed out. The theoretical results are applied to the numerical solution of parabolic and elliptic differential equations. Possible extensions as well as arising open problems are discussed in concluding remarks. Key words. Two-stage algebraic models, symmetric tridiagonal Toeplitz matrices, one-step iteration processes, matrix splitting methods, qualitative analysis, numerical solution of differential equations AMS subject classifications. 15A06, 65C20, 65M06, 65M60 1. Introduction. In this paper, we study the so-called two-stage algebraic models having the form My j+1 = Ny j + b; j = 0; 1; : : : ; y 0 is given; (1.1) where the matrices M; N 2 R n\Thetan have a special form suitable for various applied problems, and a vector b 2 R n is given. Such...