On Group Fourier Analysis and Symmetry Preserving Discretizations of PDEs

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 article 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 con- nected 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 classi- cal 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.