Spectral Element Method Part 1: Numerical Algorithms

It is well known that the fast and accurate solution of the partial differential equations (PDEs) governing geophysical fluid dynamics presents a great challenge. This is due to the complexity of both the PDEs themselves, and the initial and boundary conditions. There are several practical advantages to using a relatively new, high-accuracy numerical method, the Spectral Element Method (SEM), over currently standard methods, to compute such solutions. Put succinctly, SEM combines the accuracy of conventional spectral methods and the geometric flexibility of finite element methods. Figure 1 shows the gnomonic projection to the sphere from the cube tiled with 6×4² elements, yielding a quasi-uniform discretization. This paper..