On the accuracy of the numerical solution of the Dirichlet problem by finite differences

Abstract

This paper d erives numerical bounds for t he error, in c!'r tain closed regions, of th e differenC'e analog of the Dirichlet problem. It is concerned only wi th the difference between the exact solut ion of the differen ce equation and t he solu t ion of the D irichl!'t proble m. The error bounds obtained involve quan tities which can a ctually b e computed , such as t he mesh size, and t he oscillation and modulus of contin uity of t he gi ven fun ction on t he boundary. So far a s t he m ethod is concerned , t h e chief novelty is t h e use of the difference analogs of harmon ic mea sure and t he Schwarz Alternatin g Process.

Lipschitz Conditions for Harmonic and Discrete Harmonic Functions

“…We remark that if (x, y) E Rio then each of the four points (x ± h, y), (x, y ± h) belongs to R" + Sh. The existence of a unique bounded solution of the discrete analogue of the Dirichlet problem is well known for bounded regions; and for the cases where R is the half plane: y > 0, and the semi-infinite strip: y > 0, 0 < x < 7r, it has been shown in [3].…”

“…R the relation u(x, y) E Lip a (0 < a ~ 1) on the boundary S implies u(x, y) E Lip (a/3) in R + S; if R is convex, the conclusion [3, Theorem 5.1] is u(x, y) E Lip (a/2) in R + S; if R is a rectangle [3,Theorem 5.4] the solution of the difference equation is Lip (a/2).…”

On the Degree of Convergence of Solutions of Difference Equations to the Solution of the Dirichlet Problem

“…The series in the right member of (2.4) converges for y > 0; let u*(x, y) denote the right member of (2.4). Let the functions Uk(X, V), k = 1, 2, ... , be the arithmetic means of the first k partial sums of (2.4 For the difference equation problem it is known, see for instance [7], [9] or [12], that the unique solution is…”

“…Finite bounds for these derivatives may not exist and even when they do exist are very hard to find except under very stringent assumptions on the boundary values. The only available error bounds not involving higher derivatives of the solution are the non-explicit bound given by Rosenbloom [10] and the explicit error bounds given by Wasow [13] and by Walsh and Young [12] for the case of the rectangle. In [13] very severe differentiability conditions were imposed.…”

“…In [13] very severe differentiability conditions were imposed. In [10] and [12] uniform error bounds were obtained which are, in general, much too large for a particular point. For instance in [12], the uniform error was shown to vanish of the order h2/7, where h is the mesh size, provided the boundary values satisfy a Lipschitz condition of order one.…”