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Spectral methods on a triangle and W-systems

Jing Gao, Arieh Iserles

TL;DR

This work develops a stable, structure-preserving spectral framework for triangular domains using multivariate W-systems, focusing on Koornwinder-type constructions to obtain skew-Hermitian differentiation matrices. It derives explicit, recursive, and efficient differentiation matrices $X$ and $Y$, along with fast matrix-vector multiplication schemes, enabling stable time-dependent PDE discretisations on triangles. Convergence analysis shows spectral rates for analytic data under zero Dirichlet boundary conditions, and the paper discusses boundary lifting to handle general Dirichlet data and potential extensions to spectral elements. The approach promises rapid convergence, fast computation, and preservation of PDE structure in triangle-based spectral methods.

Abstract

We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we introduce a Koornwinder W-system. Once discretised by this W-system, the resulting spatial differentiation matrix is skew symmetric, affording important advantages insofar as stability and conservation of structure are concerned. We analyse the construction of the differentiation matrix and matrix vector multiplication, demonstrating optimal computational cost. Numerical convergence is illustrated through experiments with different parameter choices. As a result, our method exhibits key characteristics of a practical spectral method, inclusive of rapid convergence, fast computation and the preservation of structure of the underlying partial differential equation.

Spectral methods on a triangle and W-systems

TL;DR

This work develops a stable, structure-preserving spectral framework for triangular domains using multivariate W-systems, focusing on Koornwinder-type constructions to obtain skew-Hermitian differentiation matrices. It derives explicit, recursive, and efficient differentiation matrices and , along with fast matrix-vector multiplication schemes, enabling stable time-dependent PDE discretisations on triangles. Convergence analysis shows spectral rates for analytic data under zero Dirichlet boundary conditions, and the paper discusses boundary lifting to handle general Dirichlet data and potential extensions to spectral elements. The approach promises rapid convergence, fast computation, and preservation of PDE structure in triangle-based spectral methods.

Abstract

We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we introduce a Koornwinder W-system. Once discretised by this W-system, the resulting spatial differentiation matrix is skew symmetric, affording important advantages insofar as stability and conservation of structure are concerned. We analyse the construction of the differentiation matrix and matrix vector multiplication, demonstrating optimal computational cost. Numerical convergence is illustrated through experiments with different parameter choices. As a result, our method exhibits key characteristics of a practical spectral method, inclusive of rapid convergence, fast computation and the preservation of structure of the underlying partial differential equation.
Paper Structure (10 sections, 2 theorems, 88 equations, 4 figures)

This paper contains 10 sections, 2 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. W-systems on a triangle
  3. A W-system on the triangle
  4. A Koornwinder-type W-system and its differentiation matrices
  5. Explicit expressions for differentiation matrices
  6. A recursive relation
  7. Powers of $\mathcal{X}$ and $\mathcal{Y}$
  8. Matrix-vector multiplication of the differentiation matrices
  9. Convergence of Koornwinder-type W-functions on a triangle
  10. General Dirichlet boundary conditions

Key Result

Theorem 2.1

\newlabelth:10 Let the domain $\Omega$ be simply connected and assume that the weight function vanishes along $\partial\Omega$. Then the differentiation matrices $\mathcal{X}$ and $\mathcal{Y}$ are skew-symmetric.

Figures (4)

  • Figure 1: The coefficients $f_{n, k}$, drawn to logarithmic scale (left) and the $\ell_\infty$ (solid) and $\ell_2$ (dotted) errors (right) for $f(x, y) = {\mathrm e}^{x_1-2x_2} \sqrt{x_1x_2(1-x_1-x_2)}$, $\alpha=\beta=\gamma=1$, $N=1, \cdots, 45$.
  • Figure 2: Pointwise errors $e_N(x, y)$ for approximating $f(x, y) = {\mathrm e}^{x_1-2x_2} \sqrt{x_1x_2(1-x_1-x_2)}$, $\alpha=\beta=\gamma=1$ with $N=5$ (top left), $N=20$ (top right),$N=35$ (bottom left), $N=45$ (bottom right) respectively.
  • Figure 3: The coefficients $f_{n, k}$ scaled by $N$ (the left) and by $N^2$ (the middle) are plotted. The errors $e_\infty$ (solid) and $e_2$ (dotted) occur in the right figure for $f(x, y) = {\mathrm e}^{x_1-2x_2} \sqrt{x_1x_2(1-x_1-x_2)}$, $\alpha=\beta=\gamma=2$ and $N=1, \cdots, 45$.
  • Figure 4: The coefficients $f_{n, k}$ are shown in the left column, and the errors $\ell_\infty$ (solid), $\ell_2$ (dotted) errors (the right) are in the right column, for $f(x, y) = x(1-{\mathrm e}^y)\sin(\pi(1-x-y))$, $\alpha=\beta=\gamma=2$ by the W-system (the top row) and the orthogonal polynomial (the bottom row).

Theorems & Definitions (4)

  • Theorem 2.1
  • Proof 1
  • Lemma 3.1
  • Proof 2