Grid analogs of two functions harmonic on the plane with a cut
I. G. Belukhina · Moscow University Computational Mathematics and Cybernetics · 2007
For the Laplace equation in an unbounded domain (in the first quadrant, upper half-plane, plane with a cut), the Dirichlet and Neumann problems whose solutions are the imaginary and real parts of the complex function z 2lnz, respectively, are considered. Both problems are approximated on a square grid using the classic five-point difference scheme. The grid Fourier transform is applied to represent the solutions to the aforementioned grid problems in an integral form and obtain their asymptotic decompositions. It follows from the results that the accuracy of these grid solutions in the L ∞ norm is O(h 2), where h is the grid step.