On the truncation error in the solution of Laplaces equation by finite differences

Abstract

The difference between the solution of the Laplace differential equation and t he Laplace difference equation , d efin ed in th e same rectangle and assuming t he same boundary values , is estimated under th e assumption t hat the boundary fun ction possesses a bounded third derivative. The bound obtained is of the order of magni t ude of the square of the m esh length .

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

“…Thus following [15], we obtain from lemmas 3.1, 3.2, 3.3, and the mean value theorem By (3. 20) and lemma 3.6 we have …”

“…Wasow [15] has obtained error bounds for the rectangle which are applicable when the boundary values lpresented to t he American Mathematical Society, September 1951; abstract publisbed in Bu!. Am.…”

“…Since then, however, a paper by Wasow [15] has appeared which contains a stronger result for the case 8=3 than the original form of theorem 3.1. A combination of his methods and the original ones now yields an improvement (the present theorem 3.1) of Wasow's result in this case since we allow g (3) (x) and g (2) (x) to fail to exist for a finite number of points.…”

Convergence Estimates for Essentially Positive Type Discrete Dirichlet Problems

“…Bahvalov used his error bounds to estimate the number of arithmetic operations needed to obtain u to a prescribed accuracy. Related results were also obtained in special cases by Wasow [13], Laasonen [8], and by Volkov, cf. [11], [12], and references.…”

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

“…Furthermore It satIsfies the condItIon that the dIagonal elements are all posItive and the off-dIagonal elements are nonPOSItIve (2.10). Following this, others (3), (8), (9), (12), (20), (21) have extended the results of Gerschgorm within the framework of these condItIOns.…”