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 Papers 		Beth  A.  Wingate
Global spectral methods give exponential convergence rates and have high accuracy for smooth solutions, 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. The usual implementation on triangles becomes ill-conditioned for degree (maximum degree of the polynomial) greater than about seven. One can use preconditioning to partially improve this. The ill-conditioning of the triangle methods is particularly disconcerting because the finite element method on quadrilaterals doesn't have this problem. On the sphere, we have used up to degree 50 per element without conditioning problems. However, it is difficult to generate grids for quadrilateral elements, especially for the complex topography found in the ocean and on the surface of the Earth Also, automatic triangulation programs generate high-quality meshes with a smaller amount of labor. The other important issue is that in the quadrilateral method, one may obtain a globally diagonal mass matrix, making the calculations very efficient and fast.

Fekete Methods

[Beth Wingate]Continued work on this subject with Mark Taylor has shown a new way to obtain a globally diagonal mass matrix (without any preconditioning) and an exponentially converging method, for general domains. The first piece of work, The natural function space for triangular and tetrahedral spectral elements, shows that the Koornwinder basis, a set of multi-dimensional orthogonal polynomials, are the eigenfunctions of a Sturm-Liouville problem. Indeed these polynomials are the Legendre polynomials in the simplex, and can be generalized to Chebyshevs, or other weights. In this paper on functional spaces for triangles and tetrahedra, we also give the eigenvalues for an n-dimensional simplex. The second paper, The Fekete collocation points for triangular spectral element methods, shows that using this basis with the optimal points, the Fekete points, one may get well behaved Cardinal Functions, which means we get a globally diagonal mass matrix for explicit methods and an exponentially converging method. In this paper we show how to compute the points, list the points for up to degree 19, and give error estimates for the numerical method. The third paper, A generalized diagonal mass matrix spectral element method for non-quadrilateral elements puts the theory of the first to papers into one general format, showing how you use these two pieces to form a new method. The fourth paper, A Fekete point triangular spectral element method about the implementation of the method, is in progress.

Dubiner Methods
One choice is to use the Dubiner modified basis, based on the Koornwinder polynomials. This approach has been developed and used extensively by Sherwin and Karniadakis. It is much more expensive for explicit methods than their quadrilateral counterparts, but has the advantage of being able to easily do h-p grid adaption. We attempted in Triangular Spectral Element Methods for Geophysical Fluid Dynamics Applications to narrow this gap by deriving a new triangle basis, the interior-orthogonal basis. While there is some marginal improvement in CPU time to invert the mass matrix for moderate degrees, it still isn't competitive with the quadrilateral methods in speed and memory requirements.