Beth A. Wingate

An Algorithm for Computing Fekete Points in the Triangle

On the line and its tensor products, Fekete points are known to be the Gauss--Lobatto quadrature points. But unlike high-order quadrature, Fekete points generalize to non-tensor-product domains such as the triangle. Thus Fekete points might serve as an alternative to the Gauss--Lobatto points for certain applications. In this work we present a new algorithm to compute Fekete points and give results up to degree 19 for the triangle. For degree d > 10 these points have the smallest Lebesgue constant currently known. The computations validate a conjecture of Bos [ J. Approx. Theory, 64 (1991), pp. 271--280] that Fekete points along the boundary of the triangle are the one-dimensional Gauss--Lobatto points.

WAVE PROPAGATION IN 2-D ELASTIC MEDIA USING A SPECTRAL ELEMENT METHOD WITH TRIANGLES AND QUADRANGLES

We apply a spectral element method based upon a conforming mesh of quadrangles and triangles to the problem of 2-D elastic wave propagation. The method retains the advantages of classical spectral element methods based upon quadrangles only. It makes use of the classical Gauss–Lobatto–Legendre formulation on the quadrangles, while discretization on the triangles is based upon interpolation at the Fekete points. We obtain a global diagonal mass matrix which allows us to keep the explicit structure of classical spectral element solvers. We demonstrate the accuracy and efficiency of the method by comparing results obtained for pure quadrangle meshes with those obtained using mixed quadrangle-triangle and triangle-only meshes.

A generalized diagonal mass matrix spectral element method for non-quadrilateral elements

Abstract We introduce a Fekete point spectral element method. This is a generalization of the traditional quadrilateral based spectral element method to any general element such as triangles. It retains the exponential convergence and the diagonal mass matrix of the original method. We first solve a Sturm–Liouville problem in the square and the triangle to determine the correct functional space used for approximation. Once the functional space is known, we use the Fekete criterion to compute near optimal grids for these spaces which have the same number of points as the dimension of the functional space. This allows the construction of a well-behaved cardinal function basis which leads to a diagonal mass matrix.

Tensor product Gauss-Lobatto points are Fekete points for the cube

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the n-dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral methods based on Gauss-Lobatto points to non-tensor-product domains, since Fekete points are known to exist and have been computed in the triangle and tetrahedron. In one dimension this result was proved by Fejer in 1932, but the extension to higher dimensions in non-trivial.

A Cardinal Function Algorithm for Computing Multivariate Quadrature Points

We present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space. An effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights. Symmetry conditions are often used to reduce the number of equations and unknowns. Our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

N

The LANS-α and Leray turbulence parameterizations in primitive equation ocean modeling

be equal to the dimension of a prescribed lower dimensional polynomial space. The cardinal functions allow us to treat the quadrature weights as dependent variables and remove them, as well as an equivalent number of equations, from the numerical optimization procedure. We give results for the triangle, where for all degrees

A better, more discriminating test problem for ocean tracer transport

d \le 25

Lateral Mixing in the Eddying Regime and a New Broad‐Ranging Formulation

, we find quadrature formulas of this form which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.

Baroclinic Instabilities of the Two-Layer Quasigeostrophic Alpha Model

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time of a highly oscillatory nature. The algorithm combines the parareal method---a parallel-in-time scheme introduced in [J.-L. Lions, Y. Maday, and G. Turinici, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 661--668]---with techniques from the heterogeneous multiscale method (cf. [W. E and B. Engquist, Notices Amer. Math. Soc., 50 (2003), pp. 1062--1070]), which make use of the slow asymptotic structure of the equations [A. J. Majda and P. Embid, Theoret. Comput. Fluid Dyn., 11 (1998), pp. 155--169]. We present error bounds, based on the analysis in [M. J. Gander and E. Hairer, in Domain Decomposition Methods in Science and Engineering XVII, Springer, Berlin, 2008, pp. 45--56] and [G. Bal, in Domain Decomposition Methods in Science and Engineering, Springer, Berlin, 2005, pp. 425--432], that demonstrate convergence of the method. A complexity analysis also demonstrates that the parallel speedup increases ...