Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cheap and stable quadrature on polyhedral elements

Alvise Sommariva, Marco Vianello

TL;DR

The paper develops a tetrahedra-free, stable quadrature for polynomials on polyhedral elements by combining hyperinterpolation in a bounding box with divergence-theorem–based generation of Chebyshev moments. The final weights are computed via a matrix-vector product $w = diag(u) V m$, with $m_j = ∫_Ω φ_j(P) dP$ derived efficiently from the polyhedron's faces, and stability is guaranteed in the limit $n→∞$ by $∑|w_i| → vol(Ω)$. This yields a cheap, conditioning-free rule that remains stable even with negative weights and is practical for large-scale polyhedral FEM where sub-tessellation is undesirable. Numerical tests show machine-precision accuracy for a range of polyhedra and degrees, with costs substantially lower than comparable positive-weight, tetrahedra-free methods, highlighting its potential for fast and robust assembly of mass and stiffness matrices in polyhedral discretizations.

Abstract

We discuss a cheap tetrahedra-free approach to the numerical integration of polynomials on polyhedral elements, based on hyperinterpolation in a bounding box and Chebyshev moment computation via the divergence theorem. No conditioning issues arise, since no matrix factorization or inversion is needed. The resulting quadrature formula is theoretically stable even in the presence of some negative weights.

Cheap and stable quadrature on polyhedral elements

TL;DR

The paper develops a tetrahedra-free, stable quadrature for polynomials on polyhedral elements by combining hyperinterpolation in a bounding box with divergence-theorem–based generation of Chebyshev moments. The final weights are computed via a matrix-vector product , with derived efficiently from the polyhedron's faces, and stability is guaranteed in the limit by . This yields a cheap, conditioning-free rule that remains stable even with negative weights and is practical for large-scale polyhedral FEM where sub-tessellation is undesirable. Numerical tests show machine-precision accuracy for a range of polyhedra and degrees, with costs substantially lower than comparable positive-weight, tetrahedra-free methods, highlighting its potential for fast and robust assembly of mass and stiffness matrices in polyhedral discretizations.

Abstract

We discuss a cheap tetrahedra-free approach to the numerical integration of polynomials on polyhedral elements, based on hyperinterpolation in a bounding box and Chebyshev moment computation via the divergence theorem. No conditioning issues arise, since no matrix factorization or inversion is needed. The resulting quadrature formula is theoretically stable even in the presence of some negative weights.

Paper Structure

This paper contains 4 sections, 1 theorem, 10 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Polyhedral quadrature by hyperinterpolation
  3. Implementation and numerical tests
  4. Conclusions

Key Result

Theorem 2.1

Let $K\subset \mathbb{R}^d$ be a compact subset, $\mu$ an absolutely continuous measure on $K$ with respect to the Lebesgue measure. Denote by $\{\phi_j\}_{1\leq j\leq N}$ an orthonormal polynomial basis of $\mathbb{P}_n^d$ for $\mu$. Moreover, let $(X,\mathbf{u})=(\{P_i\},\{u_i)\})$, $1\leq i\leq \ where the quadrature weights $\{w_i\}$ are defined by the product-like moments Moreover, the formu

Figures (2)

  • Figure 1: Examples of polyhedral domains. Left: $\Omega_1$ (nonconvex, 20 facets); Center: $\Omega_2$ (convex, 760 facets); Right: $\Omega_3$ (multiply connected, 20 facets).
  • Figure 2: Relative errors $E(g_k)$ of the tetrahedra-free rule over 200 polynomial integrands of the form $g_k=(a_kx+b_ky+c_kz+d_k)^n$ on the three polyhedra of Figure 1, where $a_k,b_k,c_k,d_k$ are uniform random coefficients in $[-1,1]$ and $n=4,6,8,\dots,20$; the circles correspond to the average logarithmic error $\sum_{k=1}^{200}{\log(E(g_k))/200}$.

Theorems & Definitions (2)

  • Theorem 2.1
  • Remark 2.1