Corner cases of the tau method: symmetrically imposing boundary conditions on hypercubes

TL;DR

This paper introduces a generalized tau method for imposing boundary conditions on hypercubes, enabling accurate spectral solutions for elliptic PDEs with complex boundary conditions at shared edges and corners.

Contribution

It develops a systematic approach for boundary condition imposition in spectral methods on hypercubes, handling multiple boundary types and intersections with symmetry and compatibility.

Findings

Method accurately imposes boundary conditions on hypercubes.

Compatible boundary data are uniquely determined by tau corrections.

Approach extends to various elliptic equations and higher dimensions.

Abstract

Polynomial spectral methods produce fast, accurate, and flexible solvers for broad ranges of PDEs with one bounded dimension, where the incorporation of general boundary conditions is well understood. However, automating extensions to domains with multiple bounded dimensions is challenging because of difficulties in imposing boundary conditions at shared edges and corners. Past work has included various workarounds, such as the anisotropic inclusion of partial boundary data at shared edges or approaches that only work for specific boundary conditions. Here we present a general system for imposing boundary conditions for elliptic equations on hypercubes. We take an approach based on the generalized tau method, which allows for a wide range of boundary conditions for many different spectral schemes. The generalized tau method has the distinct advantage that the specified polynomial residual determines the exact algebraic solution; afterwards, any stable numerical scheme will find the same result. We can, therefore, provide one-to-one comparisons to traditional collocation and Galerkin methods within the tau framework. 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. Our approach works with general boundary conditions that commute on intersecting subsurfaces, including Dirichlet, Neumann, Robin, and any combination of these on all boundaries. The boundary tau corrections can be made hyperoctahedrally symmetric and easily incorporated into existing solvers. We present the method explicitly for the Poisson equation in two and three dimensions and describe its extension to arbitrary elliptic equations (e.g. biharmonic) in any dimension.