High-Order Quadrature for Implicitly Defined Surfaces and Regions

Abstract

We propose a high-order quadrature method for integrating over curves, surfaces, and enclosed regions defined implicitly as zero level sets of smooth functions. The method combines a mesh adjustment procedure with local change-of-variables parametrizations on cut elements. The mesh adjustment moves background mesh vertices away from the interface, ensuring a simple and nondegenerate intersection pattern between the level set and the simplicial mesh. On each cut element, the interface is parametrized by solving one-dimensional nonlinear equations along prescribed rays, reducing surface and region integration to standard Gauss--Legendre quadrature on fixed reference domains. The resulting curve and surface quadrature rules are local, nonrecursive, and have strictly positive weights. We prove high-order accuracy for curve, surface, and region integrals in two and three dimensions under natural smoothness and mesh consistency assumptions. Numerical experiments on representative implicit geometries confirm the predicted convergence rates.

35 pages, 10 figures; v2: substantially revised manuscript with expanded theoretical analysis and numerical experiments; title and author list updated