Application of Fourier- and Finite Element- Method to Elliptic Equations on the spherical surface

Finite element method in longitude and Fourier method in latitude are combined to solve elliptic equations on the spherical surface. Two dimensional gridded data with equiangular distances are represented Fourier series in the zonal direction, and the zonal Fourier coefficients are represented with piecewise linear functions in meridional direction. Elliptic equations such as the Poisson's equation and the Helmholtz equations are discretized with Fourier Finite Element Method (FFEM). Pole singularity associated with the metric term of inverse cosine of latitude was eliminated by applying the pole conditions that the Fourier coefficients other than zero (zonal mean component) vanish at poles. The method was tested with inversion and forward operation of elliptic equations for both scalar and vector fields. The accuracy was found far better finite difference method but inferior to spectral method. Application of the method to shallow water equations and high-order elliptic equations will be presented, and preliminary results of comparison with other methods will be discussed.