Abstract
This paper presents a new result concerning the perturbation theory of M-matrices. We give the proof of a theorem showing that some perturbations of irreducibly diagonally dominant M-matrices are monotone, together with an explicit bound of the norm of the perturbation. One of the assumptions concerning the perturbation matrix is that the sum of the entries of each of its row is nonnegative. The resulting matrix is shown to be monotone, although it may not be diagonally dominant and its off diagonal part may have some positive entries. We give as an application the proof of the second order convergence of an non-centered finite difference scheme applied to an elliptic boundary value problem.
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References
Alfa A.S., Xue J., Ye Q. (2002) Entrywise perturbation theory for diagonally dominant M-matrices with applications. Numer. Math. 90, 401–414
Bouchon, F., Peichl, G.H.: A second order immersed interface technique for an elliptic Neumann problem. Numer. Methods PDE DOI 10.1002/num.20187 (to appear)
Bramble J.H., Hubbard B.E. (1964) A finite difference analog of the Neumann problem for Poisson’s equation. SIAM J. Numer. Anal. 2, 1–14
Bramble J.H., Hubbard B.E. (1964) On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type. J. Math. Phys. 43, 117–132
Elsner L. (1997) Bounds for determinants of perturbed M-matrices. Linear Alg. Appl. 257, 283–288
Hackbusch W. (1992) Elliptic Differential Equations. Springer, Berlin Heidelberg New York
LeVeque R.J., Li Z. (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31: 1019–1044
Lorenz J. (1977) Zur inversmonotonie diskreter probleme. Numer. Math. 27, 227–238
Matsunaga N., Yamamoto T. (2000) Superconvergence of the Shortley–Weller approximation for Dirichlet problems. J. Comp. Appl. Math. 116, 263–273
Thomée V. (2001) From finite differences to finite elements. A short history of numerical analysis of partial differential equations. J. Comp. Appl. Math. 128, 1–54
van Linde H.J. (1974) High-order finite difference methods for Poisson’s equation. Math. Comput. 28, 369–391
Varga R.S. (2000) Matrix iterative analysis. Springer, Berlin Heidelberg New York
Wiegmann A., Bube K.P. (2000) The explicit-jump immersed interface method: finite difference methods for pdes with piecewise smooth solutions. SIAM J. Numer. Anal. 37, 827–862
Xue J., Jiang E. (1995) Entrywise relative perturbation theory for nonsingular M-matrices and applications. BIT 35, 417–427
Zafarullah A. (1969) Finite Difference Scheme for a third boundary value problem. J. ACM 16, 585–591
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Bouchon, F. Monotonicity of some perturbations of irreducibly diagonally dominant M-matrices. Numer. Math. 105, 591–601 (2007). https://doi.org/10.1007/s00211-006-0048-8
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DOI: https://doi.org/10.1007/s00211-006-0048-8