High-order integration on regular triangulated manifolds reaches super-algebraic approximation rates through cubical re-parameterizations
Gentian Zavalani, Oliver Sander, Michael Hecht
TL;DR
This work develops high-order volume elements (HOVE) for accurate surface integration on regular embedded manifolds by reparametrizing triangulations to cubical meshes through square-squeezing and interpolating both the geometry and the integrand with Chebyshev–Lobatto grids. A rigorous error theory ties the integration accuracy to the $r$-th total variation of the integrand and geometry, yielding super-algebraic to exponential convergence as interpolation degrees increase, independent of mesh size. Numerically, HOVE demonstrates superior accuracy and stability compared to Duffy-based and DCG methods across sphere/torus geometries, spherical harmonics, Gauss curvature, and near-singular geometries, with p-refinement enabling high-variance integrals to reach machine precision. The approach is particularly appealing for smooth integrals on manifolds and shows promise for extensions to implicit surfaces and spectral PDE solvers on surfaces. Overall, HOVE provides a theoretically grounded, practically effective route to ultra-accurate manifold integration through cubical re-parameterizations and high-order quadratures.
Abstract
We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE we introduce square-squeezing --a homeomorphic multilinear hypercube-simplex transformation reparametrizing an initial flat triangulation of the manifold to a cubical mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparameterized mesh through interpolation in Chebyshev-Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the $r^\text{th}$-order total variation of the integrand and the surface parameterization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing $r$, the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing $p$-refinements to overcome the limitations of $h$-refinements for highly varying smooth integrals.