Corner Cases of the Tau Method: Symmetrically Imposing Boundary Conditions on Hypercubes

Abstract.

Polynomial spectral methods produce fast, accurate, and flexible solvers for a wide range of PDEs with one bounded dimension, where the incorporation of general boundary conditions is well understood. However, extending these methods to domains with multiple bounded dimensions is challenging because of difficulties imposing boundary conditions at shared edges and corners. Previous approaches have included various workarounds, such as the anisotropic inclusion of boundary data at shared edges or techniques restricted to specific boundary conditions. Here, we present a general system for imposing boundary conditions in tensor-product spectral methods for elliptic equations on hypercubes. This system accommodates general boundary conditions that commute on intersecting subsurfaces, including any combinations of Dirichlet, Neumann, and Robin conditions on all boundaries. Our approach builds on the generalized tau method, a framework that unifies many different types of spectral schemes by specifying the polynomial residuals that determine their exact algebraic solutions. This system encompasses traditional collocation, classical tau, and sparse Galerkin formulations. As an essential requirement, we add specific tau corrections to the boundary conditions, in addition to the bulk PDE, which produce a unique set of compatible boundary data at shared subsurfaces. This method allows us to systematically construct generic solvers that are fully equivariant to hyperoctahedral symmetries (rotations and reflections of the domain). We present the method explicitly for the Poisson equation in two and three dimensions and describe its extension to arbitrary elliptic equations in any dimension.

Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/kburns/corner_taus and in the supplementary materials ( corner_taus-main.zip [4.46KB]). [Formula: see text]