The Natural Function Space for the Triangle and Tetrahedra

The success of the classical formulation of the spectral element method lies in the choice of functional space, grid points, and method of integration. On the quadrilateral, the prefered space is a tensor product basis of Legendre polynomials, P 2 = fspan(x n y m)j0 m N; 0 n Ng: (1) The grid points are chosen as the Gauss-Lobatto-Legendre points, which lead to an accurate, well-behaved interpolation, and subsequently well-behaved derivatives. The choice of points is also important for coupling elements together easily. By choosing a quadrature slightly less accurate than the optimal Gaussian, one achieves a computationally eecient diagonal global mass matrix for explicit methods. Despite this eeciency, there are several drawbacks to the quadrilaterals method. First, grid generation is more diicult for quadrilaterals than triangles, especially for solution adaptive methods. Second, it is known that quadrilaterals can give degenerate algorithms if the elements are too skewed. This motivates interest in nding an eecient spectral element method with triangular subdomains. Investigators who have proposed new algorithms for spectral elements on triangles include 2], 3], and 1]. In particular, Dubiner introduced a new orthogonal basis on the triangle, a warped product of Jacobi polynomials in the space, P 2 4 = fspan(x n y m)j0 < m; n; m + n Ng: (2) This is also the favorite approximation space for the nite element method. Dubiner deened this basis and outlined how to implement the method for trianglular subdomains. The Dubiner method has been successfully applied to the incompressible Navier-Stokes equations by Sherwin and Karniadakis 4], the shallow water equations by Wingate 6], and extended to three dimensional tetrahedra by Sherwin and Karniadakis 5]. While the method is stable, with eigenvalues for the rst and second derivative matrices which are no worse than the quadrilaterals, it is complicated to implement. There is an advantage to using this method for the incompressible Navier-Stokes equations because the semi-implicit formulation is no more expensive to solve than the explicit method, and the global stiiness matrix is sparse. The eeciency of this approach has been extensively optimized and studied by Karniadakis and his collaborators. While this basis is a generalized tensor product, the method as outlined by Dubiner does not have a well-behaved set of Cardinal functions. To couple elements together with C 0 continuity one must sacriice some orthogonality, which makes the stiiness matrix non-diagonal for explicit methods. This is much more expensive …