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

This paper studies block matrices A = [Aij ] ∈ C, where every block Aij ∈ C for i, j ∈ 〈m〉 = {1, 2, . . . ,m} and Aii 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.