Monotone and Oscillation Matrices Applied to Finite Difference Approximations

Abstract

Introduction.In solving boundary value problems by finite difference methods, there are two problems which are fundamental. One is to solve the matrix equations arising from the discrete approximation to a differential equation. The second is to estimate, in terms of the mesh spacing A, the difference between the approximate solution and the exact solution (discretization error). Until recently, most of the research papers considered these problems only for finite difference approximations whose associated square matrices are M-matrices.2 This paper treats both of the problems described above for a class of difference equations whose associated matrices are not M-matrices, but belong to the more general class of monotone matrices, i.e., matrices with nonnegative inverses.After some necessary proofs and definitions from matrix theory, we study the problem of estimating discretization errors. The fundamental paper on obtaining pointwise error bounds dates back to Gershgorin [12]. He established a technique, in the framework of M-matrices, with wide applicability. 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. We continue their work by presenting an easily applied criterion for recognizing monotone matrices. The procedure we use has the additional advantage of simplifying the work necessary to obtain pointwise error bounds. Using these new tools, we study the discretization error of a very accurate finite difference approximation to a second order elliptic differential equation.Our interests then shift from estimating discretization errors of certain finite difference approximations to how one would solve the resulting system of linear equations. For one-dimensional problems, this is not a serious consideration since Gaussian elimination can be used efficiently. This is basically due to the fact that the associated matrices are band matrices of fixed widths. However, for two-dimensional problems, Gaussian elimination is quite inefficient, because the associated band matrices have widths which increase with decreasing mesh size. Therefore, we need to consider other approaches.For cases where the matrices, arising from finite difference approximations, are symmetric and positive definite, many block successive over-relaxation methods

Generalized local maximum principles for finite-difference operators

“…Using a generalized local maximum principle, we establish the global principle for that operator on a general domain. This is a slight strengthening of the results of Bramble and Hubbard [5] and Price [18], who established the global principle for mesh-lined regions only. Also, the derivation here is considerably simpler.…”

“…(i) The nine-point-cross 0(h*) finite-difference Laplace operator in a general region. This case was slightly generalized (inclusion of lower order terms) by Price [18].…”

Optimality and resonances in a class of compact finite difference schemes of high order

“…The first application of this diophantine criteria is based on a classical result about approximation of algebraic numbers by rational ones. It gives a way to design sequences of matrices D N that are stable (8), but such that D N has not a positive or a negative spectrum for all N . [9] page 17) Let (ν q ) q be a sequence of positive real numbers such that the series ν q converges.…”

Total positivity of a kind of generalized Toric-Bernstein basis

“…Total positivity (TP) is an important and powerful concept which often occurs in many subjects, such as statistics [1,2], mathematical biology [3], combinatorics [4,5], dynamics [6], approximation theory [7,8], operator theory [9], and geometry [10]. In addition, we often see TP in the areas of graph theory, algebraic geometry, stochastic process, game theory, matroid, differential equation and representation theory.…”