Информация о публикации

Просмотр записей
Инд. авторы: Guessab A., Semisalov B.V.
Заглавие: A Multivariate Version of Hammer's Inequality and Its Consequences in Numerical Integration
Библ. ссылка: Guessab A., Semisalov B.V. A Multivariate Version of Hammer's Inequality and Its Consequences in Numerical Integration // RESULTS IN MATHEMATICS. - 2018. - Vol.73. - Iss. 1. - Art.UNSP 33. - ISSN 1422-6383.
Внешние системы: DOI: 10.1007/s00025-018-0788-7; РИНЦ: 35485677; SCOPUS: 2-s2.0-85042072028; WoS: 000426765600022;
Реферат: eng: According to Hammer's inequality (Hammer in Math Mag 31: 193-195, 1958), which is a refined version of the famous Hermite-Hadamard inequality, the midpoint rule is always more accurate than the trapezoidal rule for any convex function defined on some real numbers interval [a, b]. In this paper we consider some properties of a multivariate extension of this result to an arbitrary convex polytope. The proof is based on the use of Green formula. In doing so, we will prove an inequality recently conjectured in (Guessab and Semisalov in BIT Numerical Mathematics, 2018) about a natural multivariate version of the classical trapezoidal rule. Our proof is based on a generalization of Hammer's inequality in a multivariate setting. It also provides a way to construct new "extended" cubature formulas, which give a reasonably good approximation to integrals in which they have been tested. We particularly pay attention to the explicit expressions of the best possible constants appearing in the error estimates for these cubatute formulas.
Ключевые слова: CROUZEIX-RAVIART ELEMENT; error estimates; best constants; approximation; Cubature; convexity;
Издано: 2018
Цитирование:
1. Acu, A.M., Gonska, H.: Generalized Alomari Functionals. Mediterr. J. Math. 14, 1-17 (2017)
2. Bachar, M., Guessab, A.: A simple necessary and sufficient condition for th enrichment of the Crouzeix-Raviart element. Appl. Anal. Discrete Math. 10, 378-393 (2016)
3. 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)
4. Brenner, S.C.: Forty years of the Crouzeix-Raviart element. Numer. Methods Part. Differ. Equ. 31, 367-396 (2015)
5. Crouzeix, M., Raviart, P.A.: Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numer. 7, 33-76 (1973)
6. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
7. Guessab, A., Semisalov, B.: BIT Numerical Mathematics (2018)
8. Guessab, A., Nouisser, 0., Pecaric, J.: A multivariate extension of an inequality of Brenner-Alzer. Arch. Math. (Basel) 98(3), 277-287 (2012)
9. Guessab, A., Schmeisser, G.: Convexity results and sharp error estimates in approximate multivariate integration. Math. Comput. 73(247), 1365-1384 (2004)
10. Guessab, A.: Approximations of differentiable convex functions on arbitrary convex polytopes. Appl. Math. Comput. 240, 326-338 (2014)
11. Hammer, P.C.: The midpoint method of numerical integration. Math. Mag. 31, 193-195 (1958)
12. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Texts in Applied Mathematics, vol. 54, Springer, New York (2008)
13. Mitroi, F.C., Spiridon, C.I.: Refinements of Hermite-Hadamard inequality on simplices. Math. Rep. (Bucur.) 15, 69-78 (2013)
14. 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)
15. Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentice-Hall, Englewood Cliffs (1971)
16. 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)
17. Wasowicz, S.: Hermite-Hadamard-type inequalities in the approximate integration. Math. Inequal. Appl. 11, 693-700 (2008)
18. Zlatko, P.: Improvements of the Hermite-Hadamard inequality for the simplex. J. Inequal. Appl. 2017(1), 3 (2017)