A numerical scheme based on mean value solutions for the Helmholtz equation on triangular grids

A numerical treatment for the Dirichlet boundary value problem on regular triangular grids for homogeneous Helmholtz equations is presented, which also applies to the convection-diffusion problems. The main characteristic of the method is that an accuracy estimate is provided in analytical form with a better evaluation than that obtained with the usual finite difference method. Besides, this classical method can be seen as a truncated series approximation to the proposed method. The method is developed from the analytical solutions for the Dirichlet problem on a ball together with an error evaluation of an integral on the corresponding circle, yielding O(h4) accuracy. Some numerical examples are discussed and the results are compared with other methods, with a consistent advantage to the solution obtained here.