Institution: Stanford University, Department of Computer Science

Title: "On the Properties of the Derivatives of the Solutions of Laplace's Equation and the Errors of the Method of Finite Differences for Boundary Values in $C_2$ and $C_{1,1}$" by E. A. Volkov

Author: Volkov, E. A.

Author: Forsythe, George E.

Date: January 1970

Abstract: If a function u is harmonic in a circular disk and its boundary values are twice continuously differentiable, u need not have bounded second derivatives in the open disk. For the Dirichlet problem for Laplace's equation in a more general two-dimensional region the discretization error of the ordinary method of finite differences is studied, when Collatz's method of linear interpolation is used at the boundary. If the boundary of the region has a tangent line whose angle satisfies a Lipschitz condition, and if the boundary values have a first derivative satisfying a Lipschitz condition, then the discretization error is shown to be of order $h^2 ln h^{-1}$. This bound is shown to be sharp. By a different method of interpolation at the boundary one can improve the bound to o($h^2$). There are other similar results. Translated by G. E. Forsythe.

http://i.stanford.edu/pub/cstr/reports/cs/tr/70/151/CS-TR-70-151.pdf