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A Priori and a Posteriori Error Analysis of the Crouzeix–Raviart and Morley FEM with Original and Modified Right-Hand Sides

Carsten Carstensen and Neela Nataraj EMAIL logo

Abstract

This article on nonconforming schemes for m harmonic problems simultaneously treats the Crouzeix–Raviart ( m = 1 ) and the Morley finite elements ( m = 2 ) for the original and for modified right-hand side F in the dual space V * := H - m ( Ω ) to the energy space V := H 0 m ( Ω ) . The smoother J : V nc V in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator I nc : V V nc , and modifies the discrete right-hand side F h := F J V nc * . The best-approximation property of the modified scheme from Veeser et al. (2018) is recovered and complemented with an analysis of the convergence rates in weaker Sobolev norms. Examples with oscillating data show that the original method may fail to enjoy the best-approximation property but can also be better than the modified scheme. The a posteriori analysis of this paper concerns data oscillations of various types in a class of right-hand sides F V * . The reliable error estimates involve explicit constants and can be recommended for explicit error control of the piecewise energy norm. The efficiency follows solely up to data oscillations and examples illustrate this can be problematic.

MSC 2010: 65N30; 65N12; 65N50

Dedicated to Peter Wriggers on the occasion of his seventieth birthday


Award Identifier / Grant number: CA 151/22-2

Funding statement: The research of the first author has been supported by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 under the project “foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics” (CA 151/22-2). The finalization of this paper has been supported by SPARC project (id 235) entitled the mathematics and computation of plates.

References

[1] S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea, Providence, 2010. 10.1090/chel/369Search in Google Scholar

[2] D. N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 7–32. 10.1051/m2an/1985190100071Search in Google Scholar

[3] D. N. Arnold and R. S. Falk, A uniformly accurate finite element method for the Reissner–Mindlin plate, SIAM J. Numer. Anal. 26 (1989), no. 6, 1276–1290. 10.1137/0726074Search in Google Scholar

[4] R. Becker, S. Mao and Z. Shi, A convergent nonconforming adaptive finite element method with quasi-optimal complexity, SIAM J. Numer. Anal. 47 (2010), no. 6, 4639–4659. 10.1137/070701479Search in Google Scholar

[5] L. Beirão da Veiga, J. Niiranen and R. Stenberg, A posteriori error estimates for the Morley plate bending element, Numer. Math. 106 (2007), no. 2, 165–179. 10.1007/s00211-007-0066-1Search in Google Scholar

[6] H. Blum and R. Rannacher, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci. 2 (1980), no. 4, 556–581. 10.1002/mma.1670020416Search in Google Scholar

[7] S. C. Brenner, Forty years of the Crouzeix–Raviart element, Numer. Methods Partial Differential Equations 31 (2015), no. 2, 367–396. 10.1002/num.21892Search in Google Scholar

[8] S. C. Brenner and L.-Y. Sung, C 0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput. 22/23 (2005), 83–118. 10.1007/s10915-004-4135-7Search in Google Scholar

[9] C. Carstensen, Lectures on adaptive mixed finite element methods, Mixed Finite Element Technologies, CISM Courses and Lect. 509, Springer, Vienna (2009), 1–56. 10.1007/978-3-211-99094-0_1Search in Google Scholar

[10] C. Carstensen, S. Bartels and S. Jansche, A posteriori error estimates for nonconforming finite element methods, Numer. Math. 92 (2002), no. 2, 233–256. 10.1007/s002110100378Search in Google Scholar

[11] C. Carstensen, M. Eigel, R. H. W. Hoppe and C. Löbhard, A review of unified a posteriori finite element error control, Numer. Math. Theory Methods Appl. 5 (2012), no. 4, 509–558. 10.4208/nmtma.2011.m1032Search in Google Scholar

[12] C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math. 126 (2014), no. 1, 33–51. 10.1007/s00211-013-0559-zSearch in Google Scholar

[13] C. Carstensen, D. Gallistl and J. Hu, A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles, Numer. Math. 124 (2013), no. 2, 309–335. 10.1007/s00211-012-0513-5Search in Google Scholar

[14] C. Carstensen, D. Gallistl and J. Hu, A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes, Comput. Math. Appl. 68 (2014), no. 12, 2167–2181. 10.1016/j.camwa.2014.07.019Search in Google Scholar

[15] C. Carstensen, D. Gallistl and M. Schedensack, Adaptive nonconforming Crouzeix–Raviart FEM for eigenvalue problems, Math. Comp. 84 (2015), no. 293, 1061–1087. 10.1090/S0025-5718-2014-02894-9Search in Google Scholar

[16] C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comp. 83 (2014), no. 290, 2605–2629. 10.1090/S0025-5718-2014-02833-0Search in Google Scholar

[17] C. Carstensen, J. Gedicke and D. Rim, Explicit error estimates for Courant, Crouzeix–Raviart and Raviart–Thomas finite element methods, J. Comput. Math. 30 (2012), no. 4, 337–353. 10.4208/jcm.1108-m3677Search in Google Scholar

[18] C. Carstensen and F. Hellwig, Constants in discrete Poincaré and Friedrichs inequalities and discrete quasi-interpolation, Comput. Methods Appl. Math. 18 (2018), no. 3, 433–450. 10.1515/cmam-2017-0044Search in Google Scholar

[19] C. Carstensen and J. Hu, A unifying theory of a posteriori error control for nonconforming finite element methods, Numer. Math. 107 (2007), no. 3, 473–502. 10.1007/s00211-007-0068-zSearch in Google Scholar

[20] C. Carstensen, J. Hu and A. Orlando, Framework for the a posteriori error analysis of nonconforming finite elements, SIAM J. Numer. Anal. 45 (2007), no. 1, 68–82. 10.1137/050628854Search in Google Scholar

[21] C. Carstensen, G. Mallik and N. Nataraj, Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity, IMA J. Numer. Anal. 41 (2021), no. 1, 164–205. 10.1093/imanum/drz071Search in Google Scholar

[22] C. Carstensen and N. Nataraj, Adaptive Morley FEM for the von Kármán equations with optimal convergence rates, preprint (2019), https://arxiv.org/abs/1908.08013; to appear in SIAM J. Numer. Anal. 10.1137/20M1335613Search in Google Scholar

[23] C. Carstensen and N. Nataraj, Mathematics and computation of plates, in preparation (2021). Search in Google Scholar

[24] C. Carstensen, D. Peterseim and M. Schedensack, Comparison results of finite element methods for the Poisson model problem, SIAM J. Numer. Anal. 50 (2012), no. 6, 2803–2823. 10.1137/110845707Search in Google Scholar

[25] C. Carstensen and S. Puttkammer, How to prove the discrete reliability for nonconforming finite element methods, J. Comput. Math. 38 (2020), no. 1, 142–175. 10.4208/jcm.1908-m2018-0174Search in Google Scholar

[26] C. Carstensen and S. Puttkammer, Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian, in preparation (2021). Search in Google Scholar

[27] P. Ciarlet, C. F. Dunkl and S. A. Sauter, A family of Crouzeix–Raviart finite elements in 3D, Anal. Appl. (Singap.) 16 (2018), no. 5, 649–691. 10.1142/S0219530518500070Search in Google Scholar

[28] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. 10.1115/1.3424474Search in Google Scholar

[29] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. 3, 33–75. 10.1051/m2an/197307R300331Search in Google Scholar

[30] W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch and S. A. Sauter, Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method, Math. Comp. 73 (2004), no. 247, 1107–1138. 10.1090/S0025-5718-03-01583-7Search in Google Scholar

[31] E. Dari, R. Duran, C. Padra and V. Vampa, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 4, 385–400. 10.1051/m2an/1996300403851Search in Google Scholar

[32] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. Search in Google Scholar

[33] G. B. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton University, Princeton, 1995. Search in Google Scholar

[34] D. Gallistl, Adaptive finite element computation of eigenvalues, Ph.D. thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2014. Search in Google Scholar

[35] D. Gallistl, Morley finite element method for the eigenvalues of the biharmonic operator, IMA J. Numer. Anal. 35 (2015), no. 4, 1779–1811. 10.1093/imanum/dru054Search in Google Scholar

[36] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Class. Math., Springer, Berlin, 2001. 10.1007/978-3-642-61798-0Search in Google Scholar

[37] P. Grisvard, Singularities in Boundary Value Problems, Rech. Math. Appl. 22, Masson, Paris, 1992. Search in Google Scholar

[38] T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169–2189. 10.1090/S0025-5718-10-02360-4Search in Google Scholar

[39] J. Hu and Z. Shi, A new a posteriori error estimate for the Morley element, Numer. Math. 112 (2009), no. 1, 25–40. 10.1007/s00211-008-0205-3Search in Google Scholar

[40] J. Hu, Z. Shi and J. Xu, Convergence and optimality of the adaptive Morley element method, Numer. Math. 121 (2012), no. 4, 731–752. 10.1007/s00211-012-0445-0Search in Google Scholar

[41] J. Hu and Z.-C. Shi, The best L 2 norm error estimate of lower order finite element methods for the fourth order problem, J. Comput. Math. 30 (2012), no. 5, 449–460. 10.4208/jcm.1203-m3855Search in Google Scholar

[42] T. Kato, Estimation of iterated matrices, with application to the von Neumann condition, Numer. Math. 2 (1960), 22–29. 10.1007/BF01386205Search in Google Scholar

[43] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 181, Springer, New York, 1972. 10.1007/978-3-642-65161-8Search in Google Scholar

[44] L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart. 19 (1968), 149–169. 10.1017/S0001925900004546Search in Google Scholar

[45] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris, 1967. Search in Google Scholar

[46] H. Rabus, A natural adaptive nonconforming FEM of quasi-optimal complexity, Comput. Methods Appl. Math. 10 (2010), no. 3, 315–325. 10.2478/cmam-2010-0018Search in Google Scholar

[47] D. B. Szyld, The many proofs of an identity on the norm of oblique projections, Numer. Algorithms 42 (2006), no. 3–4, 309–323. 10.1007/s11075-006-9046-2Search in Google Scholar

[48] L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007. Search in Google Scholar

[49] R. Vanselow, New results concerning the DWR method for some nonconforming FEM, Appl. Math. 57 (2012), no. 6, 551–568. 10.1007/s10492-012-0033-8Search in Google Scholar

[50] A. Veeser and P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. I—Abstract theory, SIAM J. Numer. Anal. 56 (2018), no. 3, 1621–1642. 10.1137/17M1116362Search in Google Scholar

[51] A. Veeser and P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. III—Discontinuous Galerkin and other interior penalty methods, SIAM J. Numer. Anal. 56 (2018), no. 5, 2871–2894. 10.1137/17M1151675Search in Google Scholar

[52] A. Veeser and P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. II—Overconsistency and classical nonconforming elements, SIAM J. Numer. Anal. 57 (2019), no. 1, 266–292. 10.1137/17M1151651Search in Google Scholar

[53] R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods, Numer. Math. Sci. Comput., Oxford University, Oxford, 2013. 10.1093/acprof:oso/9780199679423.001.0001Search in Google Scholar

[54] M. Wang and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numer. Math. 103 (2006), no. 1, 155–169. 10.1007/s00211-005-0662-xSearch in Google Scholar

[55] M. Wang and J. Xu, Minimal finite element spaces for 2 m -th-order partial differential equations in R n , Math. Comp. 82 (2013), no. 281, 25–43. 10.1090/S0025-5718-2012-02611-1Search in Google Scholar

Received: 2020-12-16
Accepted: 2021-02-15
Published Online: 2021-03-11
Published in Print: 2021-04-01

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