Monotonicity and discretization error estimates

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.