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Инд. авторы: Guessab A., Semisalov B.
Заглавие: Extended multidimensional integration formulas on polytope meshes
Библ. ссылка: Guessab A., Semisalov B. Extended multidimensional integration formulas on polytope meshes // SIAM Journal on Scientific Computing. - 2019. - Vol.41. - Iss. 5. - P.A3152-A3181. - ISSN 1064-8275. - EISSN 1095-7197.
Внешние системы: DOI: 10.1137/18M1234564; РИНЦ: 41693454; SCOPUS: 2-s2.0-85074712261; WoS: 000493897100018;
Реферат: eng: In this paper, we consider a general decomposition of any convex polytope P \subset \BbbR n into a set of subpolytopes \Omega i and develop methods for approximating a definite integral of a given function f over P when, rather than its values at some points, a number of integrals of f over the faces of \Omega i are only available. We present several new families of extended integration formulas that contain such integrals and provide in a special case of our result the multivariate analogues of midpoint, trapezoidal, Hammer, and Simpson rules. The paper also presents the best possible explicit constants for their approximation errors. Here we succeed in finding the connection between minimization of the global error estimate and construction of centroidal Voronoi tessellations of a given polytope with special density function depending on properties of the integrand. In the case of integrands with strong singularities, it leads to essential reduction of the error. These ideas were extended to a more general case, in which the domain is not necessary polytope and is not necessary convex. We perform numerical tests with integrands having steep gradients which allow the comparison of the new cubature formulas and show their accuracy and rates of convergence. © 2019 Society for Industrial and Applied Mathematics
Ключевые слова: Errors; Singularity of integrand; Error estimates; Cubature formula; Convexity; Centroidal Voronoi Tessellation; Approximation; Codes (symbols); Singularity of integrand; Cubature formulas; Convexity; Centroidal Voronoi tessellation; Best error estimates; Approximation; Object recognition;
Издано: 2019
Цитирование: 
1. B. Achchab, A. Agouzal, A. Guessab, and Y. Zaim, An extended family of nonconforming quasi-Wilson elements for solving elasticity problem, Appl. Math. Comput., 344/345 (2019), pp. 1-19.
2. M. Bachar and A. Guessab, Characterization of the existence of an enriched linear finite element approximation using biorthogonal systems, Results Math., 70 (2016), pp. 401-413.
3. M. Bachar and A. Guessab, A simple necessary and sufficient condition for the enrichment of the Crouzeix-Raviart element, Appl. Anal. Discrete Math., 10 (2016), pp. 378-393.
4. M. Bachar, A. Guessab, O. Mohammed, and Y. Zaim, New cubature formulas and Hermite-Hadamard type inequalities using integrals over some hyperplanes in the d-dimensional hyper-rectangle, Appl. Math. Comput., 315 (2017), pp. 347-362.
5. E. B. Chin, J. B. Lasserre, and N. Sukumar, Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra, Comput. Mech., 56 (2015), pp. 967-981.
6. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
7. M. Crouzeix and P. A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numer., 7 (1973), pp. 33-76.
8. Q. Du, V. Faber, and M. Gunzburger, Centroidal Voronoi tessellations: Applications and algorithms, SIAM Rev., 41 (1999), pp. 637-676, https://doi.org/10.1137/S0036144599352836.
9. A. Guessab and G. Schmeisser, Convexity results and sharp error estimates in approximate multivariate integration, Math. Comp., 73 (2004), pp. 1365-1384.
10. A. Guessab and B. Semisalov, A multivariate version of Hammer's inequality and its consequences in numerical integration, Results Math., 73 (2018), 33.
11. A. Guessab and B. Semisalov, Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope, BIT, 58 (2018), pp. 613-660.
12. A. Guessab and Y. Zaim, A unified and general framework for enriching finite element approximations, in Progress in Approximation Theory and Applicable Complex Analysis, Springer Optim. Appl. 117, Springer, Cham, 2017, pp. 491-519.
13. J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods. Algorithms, Analysis, and Applications, Texts Appl. Math. 54, Springer-Verlag, New York, 2008.
14. J. B. Lasserre, Integration on a convex polytope, Proc. Amer. Math. Soc., 12 (1998), pp. 2433-2441.
15. A. Ouazzi and M. Turek, Unified edge-oriented stabilization of nonconforming FEM for incompressible flow problems: Numerical investigations, J. Numer. Math., 15 (2007), pp. 299-322.
16. C. Talischi, G. Paulino, A. Pereira, and I. F. M. Menezes, Polymesher: A general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidisc. Optim., 45 (2012), pp. 309-328.