On the Truncation Error of Discrete Approximations to the Solutions of Dirichlet Problems in a Domain with Corners

The solution for a Dirichlet problem on a given plane domain and with given boundary values is usually approximated in numerical computation by its discrete analog defined and determined on an approximating set of net points. It can be proved that the approximation thus obtained converges to the exact solution, when the net becomes denser indefinitely, independently of the domain and the boundary values subject to rather weak conditions. Nevertheless, the irregularity of the boundary curve and of the boundary values affects strongly the convergence rate. For instance, if both are analytic and if a proper boundary interpolation scheme is used, then the simplest net analog leads to an error which decreases asymptotically at least proportional to the square of the mesh constant h , as proved by Gerschgorin [2] and Collatz [1]:δ h = O ( h 2 ).