On a Finite Difference Analogue of an Elliptic Boundary Problem which is Neither Diagonally Dominant Nor of Non‐negative Type

High-order finite-difference methods for Poisson’s equation

“…The operators by which the various differential operators are approximated will be chosen in such a way that the coefficient matrix A of the resulting system of linear equations will have a very useful property, both for estimating the discretization error and for actually solving the systems, the matrix being of positive type [6]. Generally, in problems of this type, one has to attend to three things; first one has to establish the convergence of the proposed method, then one has to show that the resulting system of linear equations, which will have a sparse coefficient matrix, can be solved by iterative methods, and, finally, the stability of the method has to be investigated.…”

Monotone and Oscillation Matrices Applied to Finite Difference Approximations

“…As will be seen later, it is much easier to find a monotone matrix M which dominates A, giving a nonnegative R, than to choose R such that property 2 of Theorem 2.2 is satisfied. This is one of the major deviations between this development and Bramble and Hubbard's in [4], [5]. Also, for this reason, we shall, from now on, be concerned with constructing the matrix M rather than the matrix R.…”

“…Many others, Batschelet [1], Collatz [6] and [7], and Forsythe and Wasow [9], to name a few, have generalized Gershgorin's basic work, but their methods still used only M-matrices. Recently, Bramble and Hubbard [4] and [5] considered a class of finite difference approximations without the M-matrix sign property, except for points adjacent to the boundary. They established a technique for recognizing monotone matrices and extended Gershgorin's work to a whole class of high order difference approximations whose associated matrices were monotone rather than M-matrices.…”

“…However, the proof given here is substantially different from theirs, and gives a computable sufficient upper bound for A, (3.35). 4. Extension to a General Bounded Region.…”

The Exact Order of Convergence for Finite Difference Approximations to Ordinary Boundary Value Problems

“…In order to obtain this rule it is evident that instead of the usual stability inequalities one needs weighted stability inequalities (see Section 2). These are derived in [4], [5] from the monotone properties of the difference equations and from the existence of appropriate grid functions (for a systematic approach see Ciarlet [6]). …”