Monotonicity and Discretization Error Estimates
Owe Axelsson, L. Yu. Kolotilina · SIAM Journal on Numerical Analysis · 1990
Precise bounds of supremum norms are given for the inverse of monotone matrices. This result sharpens an earlier result by Varga [“Matrix Iterative Analysis,” Prentice–Hall, Englewood Cliffs, NJ, 1962] for H-matrices that are monotone. Various algebraic approaches to prove monotonicity based on an extended version of the weak regular splitting theorem are presented and applied to some important examples of finite-difference and finite-element matrices. This, together with the above-mentioned bounds of the norm of the inverse operator, shows that the discretization error in supremum and least-square norms for self-adjoint problems can be bounded by a constant times the norm of the truncation error, where the best constant is available.