How to recover the gradient of linear elements on nonuniform triangulations

Hlaváček, Ivan ; Křížek, Michal ; Pištora, Vladislav

Applications of Mathematics, Tome 41 (1996) no. 4, pp. 241-267

Abstract (VO)

We propose and examine a simple averaging formula for the gradient of linear finite elements in $R^d$ whose interpolation order in the $L^q$-norm is $\mathcal O(h^2)$ for $d2q$ and nonuniform triangulations. For elliptic problems in $R^2$ we derive an interior superconvergence for the averaged gradient over quasiuniform triangulations. A numerical example is presented.

Détail

DOI : 10.21136/AM.1996.134325

Classification : 65N15, 65N30
Keywords: weighted averaged gradient; linear elements; nonuniform triangulations; superapproximation; superconvergence

Bibliographie

[1] M. Ainsworth, A. Craig: A posteriori error estimators in the finite element method. Numer. Math. 60 (1992), 429–463. | DOI | MR

[2] I. Babuška, A. Miller: The post-processing in the finite element method, Parts I–II. Internat. J. Numer. Methods Engrg. 20 (1984), 1085–1109, 1111–1129. | DOI

[3] H.-J. Bartsch: Taschenbuch mathematischer Formeln. VEB Fachbuchverlag, Leipzig, 1979. | MR

[4] D. Begis, R. Glowinski: Applications de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Appl. Math. Optim. 2 (1975), 130–169. | DOI | MR

[5] J. H. Bramble, R. S. Hilbert: Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970), 112–124. | DOI | MR

[6] J. H. Bramble, A. H. Schatz: Estimates for spline projections. RAIRO Anal. Numér. 10 (1976), 5–37. | MR

[7] P. G. Ciarlet: Basic error estimates for elliptic problems, Handbook of Numerical Analysis II (P. G. Ciarlet, J. L. Lions eds.). North-Holland, Amsterdam, 1991. | MR

[8] R. Durán, M. A. Muschietti, R. Rodríguez: On the asymptotic exactness of error estimators for linear triangular finite elements. Numer. Math. 59 (1991), 107–127. | DOI | MR

[9] G. Goodsell, J. R. Whiteman: Pointwise superconvergence of recovered gradients for piecewise linear finite element approximations to problems of planar linear elasticity. Numer. Methods Partial Differential Equations 6 (1990), 59–74. | DOI | MR

[10] G. Goodsell, J. R. Whiteman: Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Numer. Methods Partial Differential Equations 7 (1991), 61–83. | DOI | MR

[11] E. J. Haug, K. K. Choi, V. Komkov: Design sensitivity analysis of structural systems. Academic Press, London, 1986. | MR

[12] I. Hlaváček, M. Křížek: On a superconvergent finite element scheme for elliptic systems, Parts I–III. Apl. Mat. 32 (1987), 131–154, 200–213, 276–289.

[13] I. Hlaváček, M. Křížek: Optimal interior and local error estimates of a recovered gradient of linear elements on nonuniform triangulations. J. Comput. Math (to appear). | MR

[14] V. Kantchev: Superconvergence of the gradient for linear finite elements for nonlinear elliptic problems. Proc. of the ISNA Conf., Prague, 1987, Teubner, Leipzig, 1988, 199–204. | MR | Zbl

[15] V. Kantchev, R. D. Lazarov: Superconvergence of the gradient of linear elements for 3D Poisson equation. Proc. Internat. Conf. Optimal Algorithms (ed. B. Sendov), Blagoevgrad, 1986, Izd. Bulg. Akad. Nauk, Sofia, 1986, 172–182.

[16] M. Křížek: An equilibrium finite element method in three-dimensional elasticity. Apl. Mat. 27 (1982), 46–75. | MR

[17] M. Křížek, P. Neittaanmäki: Superconvergence phenomenon arising in the finite element method from averaging gradients. Numer. Math. 45 (1984), 105–116. | DOI | MR

[18] M. Křížek, P. Neittaanmäki: On a global superconvergence of the gradient of linear triangular elements. J. Comput. Appl. Math. 18 (1987), 221–233. | DOI | MR

[19] M. Křížek, P. Neittaanmäki: On superconvergence techniques. Acta Appl. Math. 9 (1987), 175–198. | DOI | MR

[20] N. Levine: Superconvergent recovery of the gradient from piecewise linear finite-element approximations. IMA J. Numer. Anal. 5 (1985), 407–427. | DOI | MR | Zbl

[21] Q. Lin, N. Yan: A rectangle test for singular solution with irregular meshes. Proc. of Systems Sci. & Systems Engrg., Great Wall (H.K.) Culture Publ. Co., 1991, 236–237.

[22] Q. Lin, Q. Zhu: Asymptotic expansion for the derivative of finite elements. J. Comput. Math. 2 (1984), 361–363. | MR

[23] A. Louis: Acceleration of convergence for finite element solutions of the Poisson equation. Numer. Math. 33 (1979), 43–53. | DOI | MR | Zbl

[24] L. A. Oganesjan, L. A. Ruchovec: An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with a smooth boundary (Russian). Ž. Vyčisl. Mat. i Mat. Fiz. 9 (1969), 1102–1120. | MR

[25] V. Thomée, J. Xu, N. Zhang: Superconvergence of the gradient in piecewise linear finite element approximation to a parabolic problem. SIAM J. Numer. Anal. 26 (1989), 553–573. | DOI | MR

[26] L. B. Wahlbin: Local behavior in finite element methods, Handbook of Numerical Analysis II (P. G. Ciarlet, J. L. Lions eds.). North-Holland, Amsterdam, 1991, 353–522. | MR

[27] L. B. Wahlbin: Superconvergence in Galerkin finite element methods (Lecture notes). Cornell Univ., 1994, 1–243. | MR

[28] M. F. Wheeler, J. R. Whiteman: Superconvergent recovery of gradients on subdomains from piecewise linear finite element approximations. Numer. Methods Partial Differential Equations 3 (1987), 65–82. | DOI | MR

[29] R. Wohlgemuth: Superkonvergenz des Gradienten im Postprocessing von FiniteElemente-Methoden. Preprint Nr. 94, Tech. Univ. Chemnitz, 1989, 1–15.

[30] O. C. Zienkiewicz, J. Z. Zhu: The superconvergent patch recovery and a posteriori error estimates. Part 1, Internat. J. Numer. Methods Engrg. 33 (1992), 1331–1364. | DOI | MR