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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Triangular elements in the finite element method
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by James H. Bramble and Miloš Zlámal PDF
Math. Comp. 24 (1970), 809-820 Request permission

Abstract:

For a plane polygonal domain $\Omega$ and a corresponding (general) triangulation we define classes of functions ${p_m}(x,y)$ which are polynomials on each triangle and which are in ${C^{(m)}}(\Omega )$ and also belong to the Sobolev space $W_2^{(m + 1)}(\Omega )$. Approximation theoretic properties are proved concerning these functions. These results are then applied to the approximate solution of arbitrary-order elliptic boundary value problems by the Galerkin method. Estimates for the error are given. The case of second-order problems is discussed in conjunction with special choices of approximating polynomials.
References
  • Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
    • J. H. Argyris, I. Fried & D. W. Scharpf, "The tuba family of plate elements for the matrix displacement method," Aeronautical J. Roy. Aeronautical Soc., v. 72, 1968, pp. 618–623. K. Bell, Analysis of Thin Plates in Bending Using Triangular Finite Elements, The Technical University of Norway, Trondheim, 1968. K. Bell, "A refined triangular plate bending finite element," Internat. J. Numer. Methods in Engrg., v. 1, 1969, pp. 101–122. I. S. Berezin & N. P. Židkov, Computing Methods. Vol. I, 2nd ed., Fizmatgiz, Moscow, 1962; English transl., of 1st ed., Pergamon Press, New York, 1965. MR 30 #4372; MR 31 #1756. W. Bosshard, "Ein neues, vollverträgliches endliches Element für Plattenbiegung," Abh. Int. Verein. Brückenbau and Hochbau, Zürich, v. 28, 1968, pp. 27–40.
  • J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI 10.1137/0707006
  • Jean Céa, Approximation variationnelle des problèmes aux limites, Ann. Inst. Fourier (Grenoble) 14 (1964), no. fasc. 2, 345–444 (French). MR 174846
  • R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943), 1–23. MR 7838, DOI 10.1090/S0002-9904-1943-07818-4
    • J. J. Goël, List of Basic Functions for Numerical Utilisation of Ritz’s Method. Application to the Problem of the Plate, École Polytechnique Féderale, Lausanne, 1969.
  • Jan Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czechoslovak Math. J. 14(89) (1964), 386–393 (Russian, with English summary). MR 170088
    • J. Kratochvíl, A. Ženíšek & M. Zlámal, "A simple algorithm for the stiffness matrix of triangular plate bending finite elements," Numer. Methods in Engineering. (To appear.)
  • J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
  • J. Nitsche, Lineare Spline-Funktionen und die Methoden von Ritz für elliptische Randwertprobleme, Arch. Rational Mech. Anal. 36 (1970), 348–355 (German). MR 255043, DOI 10.1007/BF00282271
    • M. J. Turner, R. W. Clough, H. C. Martin & L. J. Topp, "Stiffness and deflection analysis of complex structures," J. Aeronautical Sci., v. 23, 1956, pp. 805–823. M. Visser, The Finite Element Method in Deformation and Heat Conduction Problems, Delft, 1968.
  • Alexander Ženíšek, Interpolation polynomials on the triangle, Numer. Math. 15 (1970), 283–296. MR 275014, DOI 10.1007/BF02165119
    • O. C. Zienkiewicz, The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, New York, 1967.
  • Miloš Zlámal, On the finite element method, Numer. Math. 12 (1968), 394–409. MR 243753, DOI 10.1007/BF02161362
    • M. Zlámal, "A finite element procedure of the second order of accuracy," Numer. Math., v. 16, 1970, pp. 394–402. A. C. Felippa, Refined Finite Element Analysis of Linear and Nonlinear TwoDimensional Structures, SESM Report No. 66–22, University of California, Berkeley, Calif., 1967. E. Anderheggen, Programme zur Methode der finiten Elemente, Institut für Baustatik, Eidgenössische Technische Hochschule, Zürich, 1969. G. R. Cowper, E. Kosko, G. M. Lindberg & M. D. Olson, "Formulation of a new triangular plate bending element," C.A.S.I. Trans., v. 1, 1968, pp. 86–90. G. R. Cowper, E. Kosko, G. M. Lindberg & M. D. Olson, "Static and dynamic applications of a high-precision triangular plate bending element," AIAA J., v. 7, 1969, pp. 1957–1965.
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Additional Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Math. Comp. 24 (1970), 809-820
  • MSC: Primary 65.66
  • DOI: https://doi.org/10.1090/S0025-5718-1970-0282540-0
  • MathSciNet review: 0282540