An extension of the class of matrices arising in the numerical solution of PDEs

This paper studies block matrices A = [A _ij] ∈ C ^km×km, where every block A _ij ∈ C ^k×k for i, j ∈ m = {1, 2,..., m} and A _ij is non-Hermitian positive definite for all i ∈ m. Such a matrix is called an extended H-matrix if its block comparison matrix is a generalized M-matrix. Matrices of this type are an extension of generalized M -matrices proposed by Elsner and Mehrmann [L. Elsner and V. Mehrmann. Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations. Numer. Math., 59:541-559, 1991.] and generalized H-matrices by Nabben [R. Nabben. On a class of matrices which arise in the numerical solution of Euler equations. Numer. Math., 63:411-431, 1992.]. This paper also discusses some properties including positive definiteness and invariance under block Gaussian elimination of a subclass of extended H-matrices, especially, convergence of some block iterative methods for linear systems with such a subclass of extended H-matrices. Furthermore, the incomplete LDU -factorization of these matrices is investigated and applied to establish some convergent results on some iterative methods. Finally, this paper generalizes theory on generalized H-matrices and answers the open problem proposed by R. Nabben.