Theory of a Finite Difference Method on Irregular Networks

A new theory is presented for the approximation of second order elliptic Dirichlet boundary value problems by a well-known finite difference scheme on irregular networks which cannot be handled by the finite element theory. Adequate regularity conditions are imposed on the network, a finite difference scheme is set up, and the convergence of its solution towards the solution of the continuous boundary value problems is established, as well as a bound for the discretization error. A numerical example illustrates the theory.