Fourier-Finite Element Method with linear basis functions on the sphere

The Fourier-Finite Element Method (FFEM) with linear basis function was applied to the partial differential equations which represent one-layer atmospheric motions on the sphere. The Fourier series is used to represent dependent field variables in zonal direction and the Fourier coefficients are expanded with series of linear finite elements by considering the pole conditions for the derivatives up to the first-order. The pole conditions for the scalar-, vector- and the nonlinear flux variables are applied to discretize the differential operators. For the Laplacian operator, the linear element was defined as a function of the sine of latitude instead of latitude, which made it possible for the metric terms to be expressed with the second-order-polynomial. The high-order Laplacian-type filter that consists of multiple Helmholtz equations was aimed at using it as a hyper-viscosity (of numerical stabilizer) and as a numerical filter as well. The FFEM was applied to the derivatives of the first- and second- order, the advection equation, deformational flow, and the shallow water equations. The conservation property of Galerkin procedure let the numerical solutions of the differential equations even in the case of the strong nonlinear flows passing over the poles be found stable and accurate. In terms of the derivatives the convergence rate of the accuracy was estimated as for the linear elements used in the present study. The high-order filter turned out to provide a quasi-uniform resolution, which does not cause any significant influence of the narrowed grid-size at poles on the time-step size associated with the CFL condition.