Solution of the Poisson equation - Comparison of the Galerkin and control-volume methods

The two-dimensional Poisson equation is solved by the finite element method which uses rectangular elements and the bilinear shape function. Two kinds of weighting functions are employed, which correspond to the Galerkin method and the control-volume method. The results for three test problems, for which exact analytical solutions are available, show that the control-volume method leads to errors that are about one-half of the errors given by the Galerkin method. Incidentally, the five-point finite difference method is found to give errors which are nearly equal to those for the Galerkin method.