Abstract
This paper is devoted to a new error analysis of nonconforming finite element methods. Compared with the classic error analysis in literature, only weak continuity, the F-E-M-Test for nonconforming finite element spaces, and basic H m regularity for exact solutions of 2m-th order elliptic problems under consideration are assumed. The analysis is motivated by ideas from a posteriori error estimates and projection average operators. One main ingredient is a novel decomposition for some key average terms on (n − 1)-dimensional faces by introducing a piecewise constant projection, which defines the generalization to more general nonconforming finite elements of the results in literature. The analysis and results herein are conjectured to apply for all nonconforming finite elements in literature.
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References
Ainsworth M, Oden J T. A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. New York: John Wiley and Sons, 2000
Arnold D N, Brezzi F. Mixed and nonconforming finite element methods implementation, postprocessing and error estimates. RAIRO Modél Math Anal Numér, 1985, 19: 7–32
Baran A, Stoyan G. Gauss-Legendre elements: A stable, higher order non-conforming finite element family. Computing, 2007, 79: 1–21
Bazeley G P, Cheung Y K, Irons B M, et al. Triangular elements in plate bending-conforming and non-conforming solutions. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics. Wright-Patterson Air Force Base, Ohio, 1965, 547–576
Beirao da Veiga L, Niiranen J, Stenberg R. A posteriori error estimates for the Morley plate bending element. Numer Math, 2007, 106: 165–179
Berger A, Scott R, Strang G. Approximate boundary conditions in the finite element method. In: Symposia Mathematica, vol. X. London: Academic Press, 1972: 295–313
Brenner S C. Poincare-Friedrichs inequalities for piecewise H1 functions. SIAM J Numer Anal, 2004, 41: 306–324
Brenner S C, Gudi T, Sung L Y. An a posteriori error estimator for a quadratic C0 interior penalty method for the biharmonic problem. IMA J Numer Anal, 2010, 30: 777–798
Chen H R, Chen S C. C0-nonconforming elements for fourth order elliptic problem. Math Numer Sin, 2013, 35: 21–30
Ciarlet P. The Finite Element Method for Elliptic Problems. Amsterdam-New York-Oxford: North-Holland, 1978
Crouzeix M, Raviart P A. Conforming and Nonconforming finite element methods for solving the stationary Stokes equations. RAIRO 7, 1973, 3: 33–76
Feng K. On the theory of discontinuous finite elements. Math Numer Sin, 1979, 1: 378–385
Fortin M, Soulie M. A nonconforming piecewise quadratic finite element on triangles. Internat J Numer Methods Engrg, 1983, 19: 505–520
Gudi T. A new error analysis for discontinuous finite element methods for linear elliptic problems. Math Comp, 2010, 79: 2169–2189
Han H. Nonconforming elements in the mixed finite element method. J Comp Math, 1984, 2: 223–233
Hu J, Shi Z C. Constrained quadrilateral nonconforming rotated Q1 element. J Comp Math, 2005, 23: 561–586
Hu J, Shi Z C. A new a posteriori error estimate for the Morley element. Numer Math, 2009, 112: 25–40
Klouček P, Li B, Luskin M. Analysis of a class of nonconforming finite elements for crystalline microstructures. Math Comp, 1996, 65: 1111–1135
Lascaux P, Lesaint P. Some nonconforming finite elements for the plate bending problem. RAIRO Anal Numer, 1975, 1: 9–53
Mao S P, Shi Z C. On the error bounds of nonconforming finite elements. Sci China Math, 2010, 53: 2917–2926
Morley L S D. The triangular equilibrium problem in the solution of plate bending problems. Aero Quart, 1968, 19: 149–169
Park C, Sheen D W. P 1 nonconforming quadrilateral finite element methods for second-order elliptic problems. SIAM J Numer Anal, 2003, 41: 624–640
Rannacher R, Turek S. Simple nonconforming quadrilateral stokes element. Numer Methods Partial Differential Equations, 1992, 8: 97–111
Shi Z C. A convergence condition for the quadrilateral Wilson element. Numer Math, 1984, 44: 349–361
Shi Z C. The F-E-M-Test for nonconforming finite elements. Math Comp, 1987, 49: 391–405
Shi Z C, Wang M. The Finite Element Method (in Chinese). Beijing: Science Press, 2010
Stummel F. The generalized patch test. SIAM J Numer Anal, 1979, 16: 449–471
Stummel F. The limitations of the patch test. Internat J Numer Methods Engrg, 1980, 15: 177–188
Verfürth R. A posteriori error estimation and adaptive mesh-refinement techniques. J Compute and Appl Math, 1994, 50: 67–83
Verfürth R. A Review of A Posteriori Error Estmation and Adaptive Mesh-Refinement Techniques. Chichester: Wiley-Teubner, 1996
Wang M. On the necessity and sufficiency of the pathc test for convergence of nonconforming finite elements. SIAM J Numer Anal, 2002, 39: 363–384
Wang M, Shi Z C, Xu J C. A new class of Zienkiewicz-type nonconforming element in any dimensions. Numer Math, 2007, 106: 335–347
Wang M, Shi Z C, Xu J C. Some n-rectangle nonconforming elements for fourth order elliptic equations. J Comp Math, 2007, 25: 408–250
Wang M, Xu J C. The Morley element for fourth order elliptic equations in any dimensions. Numer Math, 2006, 103: 155–169
Wang M, Zhang S. Nonconforming plate element with high accuracy. Research Report 70, School of Mathematical Sciences and Institute of Mathematics, Peking University, 2007
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Hu, J., Ma, R. & Shi, Z. A new a priori error estimate of nonconforming finite element methods. Sci. China Math. 57, 887–902 (2014). https://doi.org/10.1007/s11425-014-4793-3
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DOI: https://doi.org/10.1007/s11425-014-4793-3