Sparse Generalized Fourier Transforms ∗

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast. Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix. By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization pays off. The gain is significant for a direct method, and modest for an iterative method. A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is argued that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved.