Triangular Spectral Elements for Geophysical Fluid Dynamics Applications

Global spectral methods often give exponential convergence rates and have high accuracy, but are unsuitable for use with complex domains. The spectral element method combines the geometric flexibility of the finite element method with the accuracy and efficiency of the spectral method. Past efforts focus on quadrilateral elements for reasons of numerical efficiency. However, it is difficult to generate grids for quadrilateral elements, especially for the complex topography found in the ocean and atmosphere. Automatic triangulation programs generate high-quality meshes with a smaller amount of labor. In this thesis, we investigate two-dimensional spectral element methods which use triangles for their subdomains. We point out that the modified basis, derived by Dubiner, and used extensively by Sherwin and Karniadakis, is much more expensive for explicit methods than their quadrilateral counterparts. We attempt to narrow this gap by deriving a new triangle basis, the interior-orthogonal basis. The new basis results in a global stiffness matrix with diagonal interior-interior inner products that can be inverted in less CPU time. Application of the new basis to linear model problems shows that the eigenvalues of the first and second derivative matrices grow at similar rates as the modified basis. We test the robustness of the method for highly skewed triangles through numerical experimentation and show the growth of the eigenvalues is not too high. We apply the new basis to the solution of the shallow water equations on a beta plane, creating the first ocean basin model to use triangular spectral elements. The model accurately calculates the solution for the equatorially trapped Rossby solitary wave. We make further suggestions to improve model performance.