Simple formula for integration of polynomials on a simplex
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Simple formula for integration of polynomials on a simplex Jean B. Lasserre1 Received: 9 March 2020 / Accepted: 5 August 2020 © Springer Nature B.V. 2020
Abstract We show that integrating a polynomial f of degree t on an arbitrary simplex (with respect to Lebesgue measure) reduces to evaluating t homogeneous related Bombieri polynomials of degree j = 1, 2, . . . , t, each at a unique point ξ j of the simplex. This new and very simple formula could be exploited in finite (and extended finite) element methods, as well as in applications where such integrals must be evaluated. A similar result also holds for a certain class of positively homogeneous functions that are integrable on the canonical simplex. Keywords Numerical integration · Simplex · Laplace transform Mathematics Subject Classification 65D30 · 78M12 · 44A10
1 Introduction We consider the problem of integrating a polynomial f ∈ R[x] on an arbitrary simplex of Rn and with respect to the Lebesgue measure. After an affine transformation this problem is completely equivalent to integrating a related polynomial of same degree on the canonical simplex Δ = {x ∈ Rn+ : eT x ≤ 1} where e = (1, . . . , 1) ∈ Rn . Therefore the result is first proved on Δ and then transferred back to the original simplex. The result is also extended to positively homogeneous functions of the form αn α1 n α∈M f α x 1 · · · x n where M ⊂ R is finite and −1 < αi ∈ R, ∀i.
Communicated by Elisabeth Larsson. Research funded by the European Research Council (ERC) under the European’s Union Horizon 2020 research and innovation program (Grant Agreement 666981 TAMING).
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Jean B. Lasserre [email protected] LAAS-CNRS and Institute of Mathematics, University Fédérale Toulouse Midi-Pyrénées, Toulouse, France
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J. B. Lasserre
1.1 Background In addition to being a mathematical problem of independent interest, integrating a polynomial on a polytope has important applications in e.g. computational geometry, in approximation theory (constructing splines), in finite element methods, to cite a few; see e.g. the discussion in Baldoni et al. [2]. In particular, because of applications in finite (and extended finite) element methods and also for volume computation in the Natural Element Method (NEM), there has been a recent renewal of interest in developing efficient integration numerical schemes for polynomials on convex and non-convex polytopes For instance the HNI (Homogeneous Numerical Integration) technique developed in Chin et al. [4] and based on [11], has been proved to be particular efficient in some finite (and extended finite) element methods; see e.g. Antonietti et al. [1], Chin and Sukumar [5], Nagy and Benson [13] for intensive experimentation, Zhang et al. [14] for NEM, Frenning [6] for DEM (Discrete Element Method), Leoni and Shokef [12] for volume computation. For exact volume computation of polytopes in computational geometry, the interested reader is also referred to Büeler et al. [3] and references therein. For integrating a polynomial on a polytope, one possible route is to
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