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In the present paper we propose a new iterative method for the numerical solution of system of linear algebraic equations with symmetric positive definite and positive semidefinite matrices arising from the approximation of diffusion equations by the mixed hybrid finite element method on triangular and tetrahedral meshes. The method is based on a combination of the block steepest descent idea and nonoverlapping domain decomposition algorithm. We give the formulation of the method and a simple convergence estimate in the case of regular shaped quasiuniform meshes. [ABSTRACT FROM AUTHOR]

A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on a square partitioned into four squares. The collocation solution on four interfaces is computed using the preconditioned conjugate gradient method with the preconditioner defined in terms of interface preconditioners for the adjacent squares. The collocation solution on four squares is computed by a matrix decomposition method that uses fast Fourier transforms. With the number of preconditioned conjugate gradient iterations proportional to log2 N, the total cost of the algorithm is O( N2 log2 N), where the number of unknowns in the collocation solution is O( N2). The approach presented in this paper, along with that in [B.Bialecki and M.Dryja, A nonoverlapping domain decomposition method for orthogonal spline collocation problems. SIAM J. Numer. Anal. (2003) 41, 1709 – 1728.], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation. [ABSTRACT FROM AUTHOR]