A Galerkin method is applied to simple two dimensional equations important in meteorological problems. The construction of the space of trial functions for the Galerkin method is done using the “finite element” method, where the functions are defined as polynomials on individual elements and values are matched on element boundaries. This method is applied to passive advection problems and to a non-linear gravity wave problem. The results are compared with those obtained by finite difference methods and the computation time for given accuracy is shown to be at least as short using the finite element method as with finite differences. Sharp local gradients are especially well handled. Extension of this approach to irregular grids and the possible use of higher order polynomials are proposed.