Abstract
In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.
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References
Adini, A., Clough, R.W.: Analysis of plate bending by the finite element method, NSF report G. 7337 (1961)
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley Interscience, New York (2000)
Babuska, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Oxford University Press, London (2001)
Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. comp. 44, 283–301 (1985)
Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part I: Grid with superconvergence. SIAM J. Numer. Anal. 41, 2294–2312 (2003)
Baker, G.A.: Fintie element methods for elliptic equations using nonconforming elements. Math. Comp. 31, 45–59 (1977)
Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part II: general unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2003)
Brenner, S., Sung, L.: C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)
Brenner, S., Scott, L.R.: Mathematical Theory of Finite element Methods, 3rd edn. Spriger-Verlag, New York (2008)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, vol. 4. North-Holland, Amsterdam (1978)
COMSOL Multiphysics 3.5a User’s Guide, p. 471 (2008)
Introduction to COMSOL Multiphysics Version 5.1, p. 46 (2015)
Chatelin, Francoise: Spectral Approximation of Linear Operators, Computer Science and Applied Mathematics. Academic Press Inc., New York (1983)
Chen, H., Guo, H., Zhang, Z., Zou, Q.: A \(C^0\) Linear Finite Element Method For Two Fourth-Order Eigenvalue Problems IMA. J. Numer. Anal. (2016). doi:10.1093/imanum/drw051
Guo, H., Zhang, Z., Zhao, R., Zou, Q.: Polynomial preserving recovery on boundary. J. Comput. Appl. Math. 307, 119–133 (2016)
Guo, H., Yang, X.: Polynomial preserving recovery for high frequency wave propagation. J. Sci. Comput. 71, 594–614 (2017)
El-Gamel, M., Sameeh, M.: An efficient technique for finding the eigenvalues of fourth-order Sturm–Liouville problems. Appl. Math. 3, 920–925 (2012)
Lamichhane, B.: A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. J. Comput. Appl. Math. 235, 5188–5197 (2011)
Lamichhane, B.: A finite element method for a biharmonic equation based on gradient recovery operators. BIT Numer. Math. 54, 469–484 (2014)
Li, H., Nistor, V.: LNG_FEM: graded meshes on domains of polygonal structures. Rec. Adv. Sci. Comput. Appl. 586, 239–246 (2013)
Morley, L.: The triangular equilibrium problem in the solution of plate bending problems. Aero. Quart. 19, 149C–169 (1968)
Naga, A., Zhang, Z.: A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42–4, 1780–1800 (2004)
Naga, A., Zhang, Z.: The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discret. Contin. Dyn. Syst.-Ser. B 5–3, 769–798 (2005)
Naga, A., Zhang, Z.: Function value recovery and its application in eigenvalue problems. SIAM J. Numer. Anal. 50, 272–286 (2012)
Niceno, B.: EasyMesh Version 1.4: A two-dimensional quality mesh generator. http://www-dinma.univ.trieste.it/nirftc/research/easymesh
Wang, M., Xu, J.: The Morley element for fourth order elliptic equations in any dimensions. Numer. Math. 103, 155–169 (2006)
Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp. 73, 1139–1152 (2004)
Zhang, S., Zhang, Z.: Invalidity of decoupling a biharmonic equation to two Poisson equations on non-convex polygons. Int. J. Numer. Anal. Model. 5, 73–76 (2008)
Zhang, Z.: Recovery Techniques in Finite Element Methods. In: Tang, T., Xu, J. (eds.) Adaptive Computations: Theory and Algorithms. Mathematics Monograph Series 6, pp. 333–412. Science Publisher, London (2007)
Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26–4, 1192–1213 (2005)
Zienkiewicz, O.C., Zhu, J.Z.: The superconvergence patch recovery and a posteriori error estimates part 1: the recovery technique. Int. J. Numer. Methods Eng. 33, 1331–1364 (1992)
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H. Guo: The research of this author was supported in part by the US National Science Foundation through Grant DMS-1419040. Z. Zhang: The research of this author was supported in part by the following Grants: NSFC 11471031, NSFC 91430216, NASF U1530401, and NSF DMS-1419040. Q. Zou: The research of this author was supported in part by the following Grants: the special project High performance computing of National Key Research and Development Program 2016YFB0200604, NSFC 11571384, Guangdong Provincial NSF 2014A030313179, the Fundamental Research Funds for the Central Universities 16lgjc80.
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Guo, H., Zhang, Z. & Zou, Q. A \(C^0\) Linear Finite Element Method for Biharmonic Problems. J Sci Comput 74, 1397–1422 (2018). https://doi.org/10.1007/s10915-017-0501-0
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DOI: https://doi.org/10.1007/s10915-017-0501-0