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

Keaton J. Burns, Daniel Fortunato, Keith Julien, Geoffrey M. Vasil

TL;DR

This work takes an approach based on the generalized tau method, which allows for a wide range of boundary conditions for many types of spectral methods, and presents the method explicitly for the Poisson equation in two and three dimensions and describes its extension to arbitrary elliptic equations in any dimension.

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.

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

TL;DR

This work takes an approach based on the generalized tau method, which allows for a wide range of boundary conditions for many types of spectral methods, and presents the method explicitly for the Poisson equation in two and three dimensions and describes its extension to arbitrary elliptic equations in any dimension.

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.
Paper Structure (23 sections, 4 theorems, 42 equations, 9 figures)

This paper contains 23 sections, 4 theorems, 42 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The generalized tau method
  3. 2D Poisson
  4. Interior tau modifications
  5. Boundary tau modifications for Dirichlet data
  6. Collocation modifications as a tau scheme
  7. Singularity of the simplest tau modifications
  8. Nonsingular boundary tau modifications
  9. Boundary tau modifications via additional constraints
  10. Corner conditions for general commuting boundary conditions
  11. The dihedrally symmetric ultraspherical tau method
  12. 3D Poisson
  13. Interior tau corrections
  14. Corrections for general commuting boundary conditions
  15. Generalization to arbitrary dimensions and elliptic operators
  16. ...and 8 more sections

Key Result

Lemma 2.1

\newlabellemma.straight_derivatives0 Consider the differential equation $\partial^b u = f$ along with $b$-many linear boundary conditions $\mathcal{B} u = g$. When $f \in \Pi_{N-b}$, the equation has a unique solution $u \in \Pi_{N}$ if the matrix $\mathcal{B} $ is full rank.

Figures (9)

  • Figure 1: Left: We consider problems in the square with coordinates $(x,y)$ and edges labeled north ($N$), east ($E$), south ($S$), and west ($W$). Center: We consider problems in the cube with coordinates $(x,y,z)$ and faces labeled north ($N$), east ($E$), south ($S$), west ($W$), top ($T$), and bottom ($B$). Right: Standard type-II (extrema) collocation discretizations on the square contain $(N_x-2)(N_y-2)$ interior nodes (black) and $2 N_x + 2 N_y - 4$ boundary nodes (orange); all other spectral formulations of second-order elliptic problems require the same number of interior and boundary constraints.
  • Figure 1: A pictorial representation of the constraints in collocation (top) and Galerkin (bottom) tau schemes for the Poisson equation in 2D. From left to right: The interior equations are enforced on the $N-2$ interior nodes / low modes providing $(N-2)^2$ constraints. The boundary conditions on each edge are enforced on the $N-2$ interior nodes / low modes providing $4(N-2)$ constraints. Conditions on each corner provide the last $4$ constraints.
  • Figure 1: A pictorial representation of the constraints in collocation (top) and Galerkin (bottom) tau schemes for the Poisson equation in 3D. From left to right: The interior equations are enforced on the $N-2$ interior nodes / low modes providing $(N-2)^3$ constraints. The boundary conditions on each face are enforced on the $N-2$ interior nodes / low modes providing $6(N-2)^2$ constraints. Conditions on each edge are enforced on the $N-2$ interior nodes / low modes providing $12(N-2)$ constraints. Finally, conditions on each corner provide the last $8$ constraints.
  • Figure 1: Solutions to the forced 2D Poisson equation, computed using the symmetric ultraspherical tau method. The solution is computed using homogeneous Dirichlet (top left), Neumann (top right), and Robin (bottom left) boundary conditions on each edge.
  • Figure 2: Left: A naive approach to removing four constraints from the boundary conditions by placing a single tau polynomial (or dropping the highest mode) on each edge. Right: This approach is singular, because there is an unconstrained combination of these modes which is continuous at all four corners.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Lemma 2.1
  • Proof 1
  • Lemma 2.2
  • Proof 2
  • Lemma 2.3
  • Proof 3
  • Proposition 2.4
  • Proof 4
  • Conjecture 2.5