Article: On group Fourier analysis and symmetry preserving discretizations of PDEs
Author: H. Z. Munthe-Kaas
Journal: J. Phys. A: Math. Gen. 39 (2006) 5563–5584.
Abstract: In this paper we review some group theoretic techniques applied to discretizations of PDEs. Inspired by the recent years active research in
Lie group- and exponential-time integrators for differential equations, we
will in the first part of the paper present algorithms for computing matrix
exponentials based on Fourier transforms on finite groups. As an example,
we consider spherically symmetric PDEs, where the discretization preserves
the 120 symmetries of the icosahedral group. This motivates the study of
spectral element discretizations based on triangular subdivisions. In the
second part of the paper, we introduce novel applications of multivariate
non-separable Chebyshev polynomials in the construction of spectral element
bases on triangular and simplicial sub-domains. These generalized Chebyshev
polynomials are intimately connected to the theory of root systems and Weyl
groups (used in the classification of semi-simple Lie algebras), and these
polynomials share most of the remarkable properties of the classical Chebyshev
polynomials, such as near-optimal Lebesgue constants for the interpolation
error, the existence of FFT-based algorithms for computing interpolants and
pseudo-spectral differentiation and existence of Gaussian integration rules. The
two parts of the paper can be read independently.
Remarks in hindsight: None yet.
24 Apr 2006