A High Order Method for the Approximation of Integrals Over Implicitly Defined Hypersurfaces

Abstract

Abstract. We introduce a novel method to compute approximations of integrals over implicitly defined hypersurfaces. The new method is based on a weak formulation in L 2 (0, 1), that uses the coarea formula to circumvent an explicit integration over the hypersurfaces. As such it is possible to use standard quadrature rules in the spirit of hp/spectral finite element methods, and the expensive computation of explicit hypersurface parametrizations is avoided. We derive error estimates showing that high order convergence can be achieved provided the integrand and the hypersurface defining function are sufficiently smooth. The theoretical results are supplemented by numerical experiments including an application for plasma modeling in nuclear fusion.

Tokamak free-boundary plasma equilibrium computation using finite elements of class C0 and C1 within a mo

“…In [10] it is proposed to use a weak formulation based on the coarea formula (see Theorem 3.2.12 in [14]) to compute integrals such as (26). The coarea formula underlines the relationship between integrals on iso-contours and integrals on Ω ⊂ R d , here d = 2.…”

“…In [10], to approximate L 2 (0, 1), the space of all polynomials of degree less than or equal to P in [0, 1] is used as discrete space, with a basis of Legendre polynomials. This enables to have a diagonal (P + 1) × (P + 1) mass matrix corresponding to the left-hand side of ( 29) but on the other hand requires the use of an expensive high order quadrature formula to compute terms corresponding to the right-hand side of (29) since Legendre polynomials up to degree 30 or 40 have to be used.…”

“…The rst one, rather usual in equilibrium codes, relies on the explicit computation of iso-contours. The second one which proves to be very ecient is based on the coarea formula and motivated by results from [10].…”

Numerical Integration Over Implicitly Defined Domains with Topological Guarantee

“…That is, the topology could not be well preserved by the initial approximation and later corrections are made. In [33] and [34], the integration over hyperrectangles and hypersurfaces are considered, respectively.…”

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

“…These methods avoid an explicit reconstruction of the interface; instead, a sampling of the given level set function, typically on Cartesian grids, is used to approximate, e.g., Γ f = h d i f (x i )δ h (φ(x i ))|∇ h φ(x i )|, where δ h is a grid-dependent smoothed Dirac delta function and x i are the grid points. These approaches typically rely on a cancellation of errors in the summation over regularly spaced grid points and it can be subtle to develop convergent schemes [67,18,61,68,81,39], especially higher-order ones [73,74,75]; see also related methods built through application of the co-area formula [17,34,80,70].…”