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Keywords:
FEM; non-conforming linear triangle; a priori error estimate; a posteriori error estimate; error constant; Raviart-Thomas element
FEM; non-conforming linear triangle; a priori error estimate; a posteriori error estimate; error constant; Raviart-Thomas element
Summary:
The non-conforming linear ($P_1$) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuška-Aziz maximum angle condition is required just as in the case of the conforming $P_1$ triangle. Some applications and numerical results are also included to see the validity and effectiveness of our analysis.
References:
[1] Acosta, G., Durán, R. G.: The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37 (1999), 18-36. DOI 10.1137/S0036142997331293 | MR 1721268 | Zbl 0948.65115
[2] Ainsworth, M., Oden, J. T.: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics, Wiley-Interscience, New York (2000). DOI 10.1002/9781118032824 | MR 1885308 | Zbl 1008.65076
[3] Arnold, D. N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO, Modélisation Math. Anal. Numér. 19 (1985), 7-32. DOI 10.1051/m2an/1985190100071 | MR 0813687 | Zbl 0567.65078
[4] Arnold, D. N., Brezzi, F., Cockburn, B., Marini, L. D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002), 1749-1779. DOI 10.1137/S0036142901384162 | MR 1885715 | Zbl 1008.65080
[5] Babuška, I., Aziz, A. K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976), 214-226. DOI 10.1137/0713021 | MR 0455462 | Zbl 0324.65046
[6] Babuška, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford (2001). MR 1857191 | Zbl 0995.65501
[7] Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics ETH Zürich, Birkhäuser, Basel (2003). DOI 10.1007/978-3-0348-7605-6 | MR 1960405 | Zbl 1020.65058
[8] Brenner, S. C., Scott, L. R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15, Springer, Berlin (2002). DOI 10.1007/978-1-4757-3658-8 | MR 1894376 | Zbl 1012.65115
[9] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics 15, Springer, New York (1991). DOI 10.1007/978-1-4612-3172-1 | MR 1115205 | Zbl 0788.73002
[10] Carstensen, C., Gedicke, J., Rim, D.: Explicit error estimates for Courant, Crouzeix-Raviart and Raviart-Thomas finite element methods. J. Comput. Math. 30 (2012), 337-353. DOI 10.4208/jcm.1108-m3677 | MR 2965987 | Zbl 1274.65290
[11] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics 40, Society for Industrial and Applied Mathematics, Philadelphia (2002). DOI 10.1137/1.9780898719208 | MR 1930132 | Zbl 0999.65129
[12] Destuynder, P., Métivet, B.: Explicit error bounds in a conforming finite element method. Math. Comput. 68 (1999), 1379-1396. DOI 10.1090/S0025-5718-99-01093-5 | MR 1648383 | Zbl 0929.65095
[13] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Classics in Applied Mathematics 69, Society for Industrial and Applied Mathematics, Philadelphia (2011). DOI 10.1137/1.9781611972030.ch1 | MR 3396210 | Zbl 1231.35002
[14] Hu, J., Ma, R.: The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements. J. Sci. Comput. 63 (2015), 410-425. DOI 10.1007/s10915-014-9899-9 | MR 3328190 | Zbl 1319.65111
[15] Kikuchi, F.: Convergence of the ACM finite element scheme for plate bending problems. Publ. Res. Inst. Math. Sci., Kyoto Univ. 11 (1975), 247-265. DOI 10.2977/prims/1195191694 | MR 0391540 | Zbl 0326.73065
[16] Kikuchi, F., Ishii, K.: A locking-free mixed triangular element for the Reissner-Mindlin plates. S. N. Atluri, G. Yagawa, T. Cruse Computational Mechanics'95: Theory and Applications Springer, Berlin (1995), 1608-1613. DOI 10.1007/978-3-642-79654-8
[17] Kikuchi, F., Liu, X.: Determination of the Babuska-Aziz constant for the linear triangular finite element. Japan J. Ind. Appl. Math. 23 (2006), 75-82. DOI 10.1007/BF03167499 | MR 2210297 | Zbl 1098.65107
[18] Kikuchi, F., Liu, X.: Estimation of interpolation error constants for the $P_0$ and $P_1$ triangular finite elements. Comput. Methods Appl. Mech. Eng. 196 (2007), 3750-3758. DOI 10.1016/j.cma.2006.10.029 | MR 2340000 | Zbl 1173.65346
[19] Kikuchi, F., Saito, H.: Remarks on a posteriori error estimation for finite element solutions. J. Comput. Appl. Math. 199 (2007), 329-336. DOI 10.1016/j.cam.2005.07.031 | MR 2269515 | Zbl 1109.65094
[20] Knabner, P., Angermann, L.: Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Texts in Applied Mathematics 44, Springer, New York (2003). DOI 10.1007/b97419 | MR 1988268 | Zbl 1034.65086
[21] Kobayashi, K.: On the interpolation constants over triangular elements. Kyoto University Research Information Repository 1733 (2011), 58-77 Japanese.
[22] Kobayashi, K.: On the interpolation constants over triangular elements. Proceedings of the International Conference Applications of Mathematics 2015 J. Brandts et al. Czech Academy of Sciences, Institute of Mathematics, Praha (2015), 110-124. MR 3700193 | Zbl 1363.65014
[23] Liu, X., Kikuchi, F.: Estimation of error constants appearing in non-conforming linear triangular finite element. Procceding of APCOM'07 in conjunction with EPMESC XI, Kyoto (2007), Available at http://www.xfliu.org/p/2007LK.pdf\kern0pt
[24] Liu, X., Kikuchi, F.: Analysis and estimation of error constants for $P_0$ and $P_1$ interpolations over triangular finite elements. J. Math. Sci., Tokyo 17 (2010), 27-78. MR 2676659 | Zbl 1248.65118
[25] Marini, L. D.: An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22 (1985), 493-496. DOI 10.1137/0722029 | MR 0787572 | Zbl 0573.65082
[26] Nakao, M. T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optimization 22 (2001), 321-356. DOI 10.1081/NFA-100105107 | MR 1849323 | Zbl 1106.65315
[27] Nakao, M. T., Yamamoto, N.: A guaranteed bound of the optimal constant in the error estimates for linear triangular element. Topics in Numerical Analysis. With Special Emphasis on Nonlinear Problems G. Alefeld, X. Chen Comput Suppl. 15, Springer, Wien (2001), 165-173. DOI 10.1007/978-3-7091-6217-0 | MR 1874511 | Zbl 1013.65119
[28] Temam, R.: Numerical Analysis. D. Reidel Publishing Company, Dordrecht (1973). DOI 10.1007/978-94-010-2565-2 | MR 0347099 | Zbl 0261.65001
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