Инд. авторы:	Guessab A., Semisalov B.V.
Заглавие:	Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope
Библ. ссылка:	Guessab A., Semisalov B.V. Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope // BIT Numerical Mathematics. - 2018. - P.613-660. - ISSN 0006-3835. - EISSN 1572-9125.
Внешние системы:	DOI: 10.1007/s10543-018-0703-3; РИНЦ: 35701999; РИНЦ: 41774860; РИНЦ: 38606519; SCOPUS: 2-s2.0-85044368809; WoS: 000444946100006;
Реферат:	eng: In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in Rd are only available. We present several new families of “extended” integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson’s rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.
Ключевые слова:	Convexity; Best constants; Error estimates; Approximation; Cubature;
Издано:	2018
Физ. характеристика:	с.613-660
Цитирование:	1. Acu, A.M., Gonska, H.: Generalized Alomari functionals. Mediterr. J. Math. 14, 1–17 (2017) 2. Bachar, M., Guessab, A., Mohammed, O., Zaim, Y.: New cubature formulas and Hermite–Hadamard type inequalities using integrals over some hyperplanes in the d -dimensional hyper-rectangle. Appl. Math. Comput. 315(15), 347–362 (2017) 3. Bachar, M., Guessab, A.: A simple necessary and sufficient condition for the enrichment of the Crouzeix–Raviart element. Appl. Anal. Discrete Math. 10, 378–393 (2016) 4. Bachar, M., Guessab, A.: Characterization of the existence of an enriched linear finite element approximation using biorthogonal systems. Results Math. 70(3), 401–413 (2016) 5. Barvinok, A.I.: Computation of exponential integrals. J. Math. Sci. 70, 1934–1944 (1994) 6. Chin, Eric B., Lasserre, Jean B., Sukumar, N.: Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra. Comput. Mech. 56(6), 967–981 (2015) 7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) 8. Crouzeix, M., Raviart, P.A.: Conforming and non-conforming finite element methods for solving the stationary Stokes equations. R.A.I.R.O Anal. Numer. 7, 33–76 (1973) 9. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Series: Applied Mathematical Sciences, vol. 159. Springer, New York (2004) 10. Guessab, A., Schmeisser, G.: Convexity results and sharp error estimates in approximate multivariate integration. Math. Comput. 73(247), 1365–1384 (2004) 11. Guessab, A.: Approximations of differentiable convex functions on arbitrary convex polytopes. Appl. Math. Comput. 240, 326–338 (2014) 12. Hammer, P.C.: The midpoint method of numerical integration. Math. Mag. 31, 193–195 (1958) 13. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Texts in Applied Mathematics, vol. 54. Springer, New York (2008) 14. Hossain, M.A., Islam, MdSh: Generalized composite numerical integration rule over a polygon using Gaussian quadrature. Dhaka Univ. J. Sci. 62(1), 25–29 (2014) 15. Lasserre, J.B., Zeron, E.S.: Solving a class of multivariate integration problems via Laplace techniques. Appl. Math. 28(4), 391–405 (2001) 16. Ouazzi, A., Turek, S.: Unified edge-oriented stabilization of nonconforming FEM for incompressible flow problems: numerical investigations. J. Numer. Math. 15(4), 299–322 (2007) 17. Persson, P.O., Strang, G.: A simple mesh generator in MATLAB. SIAM Rev. 46, 329–345 (2004) 18. Persson, P.O.: Mesh Generation for Implicit Geometries. Ph.D. thesis, Department of Mathematics, MIT (2004) 19. Vermolen, F., Segal, G.: On an integration rule for products of barycentric coordinates over simplexes in Rd, Technical Report 17–02. Delft University of Technology, DIAM (2017)