Spectral interpolation in non-orthogonal domains: algorithms and applications

Abstract

Polynomial spectral methods have become increasingly important in diverse applications such as aeroacoustics, electromagnetics, ocean modeling, seismology, turbulent flows, non-Newtonian flows, non-linear optics, plasma dynamics, and uncertainty quantification. With more emphasis placed in the last 10 years on time-dependent rather than stationary problems, high-order methods, in general, and spectral-based methods, in particular, are potentially more effective than low-order discretizations. New developments in the last few years have addressed issues of complex geometry, finite regularity, fast solvers, multi-resolution, and large-scale simulations. The theme in the present special volume is spectral representations in non-orthogonal domains, specifically in triangles in two-dimensions and tetrahedra in three-dimensions. Multi-dimensional spectral expansions in orthogonal domains (squares or cubes) are based on tensor-products of one-dimensional expansions; for example products of Chebyshev or Legendre polynomials are popular choices. Interpolation is typically based on grid points that are given by the roots of these polynomials (Gauss points) or by the roots of the polynomials’ derivatives (Gauss–Lobatto points, if the end-points are included.) These grid points may also serve as the numerical quadrature points in spectral Galerkin formulations. However, in non-orthogonal domains, tensor-product expansion bases cannot be readily constructed, the best choice of grid points for interpolation or numerical integration is not so obvious. But good progress on both fronts has been made in the last 10 years, setting the foundation for a new generation of spectral and spectral-element methods that can handle truly complex geometries and can deal with error control very effectively. With regards to tensor products, new Jacobi-polynomial representations in conjunction with appropriate transformations have led to tensor-product bases in triangles and tetrahedra. With respect to the best set of grid points, several new algorithms have been proposed that provide excellent approximation properties in triangles and tetrahedra. Wehave collected in this special volumeeleven invited papers by leading researchers in spectralmethods. They are divided in three groups. The first group (1–7) deals with fundamental approximation properties, the second group (8–9) addresses improvements in solution procedures, and the third group (10–11) presents applications.