A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified right-hand sides. (English) Zbl 1476.65294
Summary: 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 \(V^\ast:=H^{-m}(\Omega)\) to the energy space \(V:=H^m_0(\Omega)\). The smoother \(J:V_{\mathrm{nc}}\to V\) in this paper is a companion operator, that is a linear and bounded right-inverse to the nonconforming interpolation operator \(I_{\mathrm{nc}}:V\to V_{\text{nc}}\), and modifies the discrete right-hand side \(F_h:=F\circ J\in V_{\mathrm{nc}}^\ast\). The best-approximation property of the modified scheme from A. Veeser and P. Zanotti [SIAM J. Numer. Anal. 56, No. 3, 1621–1642 (2018; Zbl 1412.65191); ibid. 56, No. 5, 2871–2894 (2018; Zbl 1447.65164); ibid. 57, No. 1, 266–292 (2019; Zbl 1414.65037)] 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 \(F\in V^\ast\). 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:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
Keywords:
nonconforming schemes; Crouzeix-Raviart; Morley; biharmonic problem; medius error analysis; best-approximation; a priori and a posteriori analysisReferences:
| [1] | S. Agmon, Lectures on Elliptic Boundary Value Problems, AMS Chelsea, Providence, 2010. · Zbl 1221.35002 |
| [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. · Zbl 0567.65078 |
| [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. · Zbl 0696.73040 |
| [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. · Zbl 1208.65154 |
| [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. · Zbl 1110.74050 |
| [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. · Zbl 0445.35023 |
| [7] | S. C. Brenner, Forty years of the Crouzeix-Raviart element, Numer. Methods Partial Differential Equations 31 (2015), no. 2, 367-396. · Zbl 1310.65142 |
| [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. · Zbl 1071.65151 |
| [9] | C. Carstensen, Lectures on adaptive mixed finite element methods, Mixed Finite Element Technologies, CISM Courses and Lect. 509, Springer, Vienna (2009), 1-56. · Zbl 1181.65136 |
| [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. · Zbl 1010.65044 |
| [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. · Zbl 1289.65249 |
| [12] | C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation, Numer. Math. 126 (2014), no. 1, 33-51. · Zbl 1298.65165 |
| [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. · Zbl 1316.74060 |
| [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. · Zbl 1362.65123 |
| [15] | C. Carstensen, D. Gallistl and M. Schedensack, Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems, Math. Comp. 84 (2015), no. 293, 1061-1087. · Zbl 1311.65136 |
| [16] | C. Carstensen and J. Gedicke, Guaranteed lower bounds for eigenvalues, Math. Comp. 83 (2014), no. 290, 2605-2629. · Zbl 1320.65162 |
| [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. · Zbl 1274.65290 |
| [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. · Zbl 1401.65130 |
| [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. · Zbl 1127.65083 |
| [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. · Zbl 1165.65072 |
| [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. · Zbl 1460.65145 |
| [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. |
| [23] | C. Carstensen and N. Nataraj, Mathematics and computation of plates, in preparation (2021). |
| [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. · Zbl 1261.65115 |
| [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. · Zbl 1463.65362 |
| [26] | C. Carstensen and S. Puttkammer, Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian, in preparation (2021). |
| [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. · Zbl 1396.33020 |
| [28] | P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. · Zbl 0383.65058 |
| [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. · Zbl 0302.65087 |
| [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. · Zbl 1050.65110 |
| [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. · Zbl 0853.65110 |
| [32] | L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. · Zbl 0902.35002 |
| [33] | G. B. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton University, Princeton, 1995. · Zbl 0841.35001 |
| [34] | D. Gallistl, Adaptive finite element computation of eigenvalues, Ph.D. thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2014. |
| [35] | D. Gallistl, Morley finite element method for the eigenvalues of the biharmonic operator, IMA J. Numer. Anal. 35 (2015), no. 4, 1779-1811. · Zbl 1332.65160 |
| [36] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Class. Math., Springer, Berlin, 2001. · Zbl 1042.35002 |
| [37] | P. Grisvard, Singularities in Boundary Value Problems, Rech. Math. Appl. 22, Masson, Paris, 1992. · Zbl 0766.35001 |
| [38] | T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp. 79 (2010), no. 272, 2169-2189. · Zbl 1201.65198 |
| [39] | J. Hu and Z. Shi, A new a posteriori error estimate for the Morley element, Numer. Math. 112 (2009), no. 1, 25-40. · Zbl 1169.74646 |
| [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. · Zbl 1255.65212 |
| [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. · Zbl 1274.65292 |
| [42] | T. Kato, Estimation of iterated matrices, with application to the von Neumann condition, Numer. Math. 2 (1960), 22-29. · Zbl 0119.32001 |
| [43] | J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 181, Springer, New York, 1972. · Zbl 0223.35039 |
| [44] | L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aero. Quart. 19 (1968), 149-169. |
| [45] | J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Paris, 1967. · Zbl 1225.35003 |
| [46] | H. Rabus, A natural adaptive nonconforming FEM of quasi-optimal complexity, Comput. Methods Appl. Math. 10 (2010), no. 3, 315-325. · Zbl 1283.65098 |
| [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. · Zbl 1102.47002 |
| [48] | L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lect. Notes Unione Mat. Ital. 3, Springer, Berlin, 2007. · Zbl 1126.46001 |
| [49] | R. Vanselow, New results concerning the DWR method for some nonconforming FEM, Appl. Math. 57 (2012), no. 6, 551-568. · Zbl 1274.65293 |
| [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. · Zbl 1412.65191 |
| [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. · Zbl 1447.65164 |
| [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. · Zbl 1414.65037 |
| [53] | R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods, Numer. Math. Sci. Comput., Oxford University, Oxford, 2013. · Zbl 1279.65127 |
| [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. · Zbl 1092.65103 |
| [55] | M. Wang and J. Xu, Minimal finite element spaces for \(2m\)-th-order partial differential equations in \(R^n\), Math. Comp. 82 (2013), no. 281, 25-43. · Zbl 1264.65199 |
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