Open Access
Issue |
ESAIM: M2AN
Volume 56, Number 1, January-February 2022
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Page(s) | 41 - 78 | |
DOI | https://doi.org/10.1051/m2an/2021085 | |
Published online | 07 February 2022 |
- S. Agmon, Lectures on Elliptic Boundary Value Problems. Providence, RI: AMS Chelsea Publishing (2010) [Google Scholar]
- G.A. Baker, Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31, (1977) 45–59 [CrossRef] [MathSciNet] [Google Scholar]
- 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) 556–581 [Google Scholar]
- D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd edn. Cambridge: Cambridge (2007) [CrossRef] [Google Scholar]
- S.C. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comp. 68, (1999) 25–53 [CrossRef] [MathSciNet] [Google Scholar]
- S.C. Brenner and L.-Y. Sung, C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22, (2005) 83–118 [CrossRef] [Google Scholar]
- S.C. Brenner, T. Gudi and L.-Y. Sung, A weakly over-penalized symmetric interior penalty method for the biharmonic problem. Electron. Trans. Numer. Anal. 37, (2010) 214–238 [MathSciNet] [Google Scholar]
- S.C. Brenner, L.-Y. Sung, H. Zhang and Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254, (2013) 31–42 [Google Scholar]
- C. Carstensen, A unifying theory of a posteriori finite element error control. Numer. Math. 100, (2005) 617–637 [Google Scholar]
- C. Carstensen and D. Gallistl, Guaranteed lower eigenvalue bounds for the biharmonic equation. Numer. Math. 126, (2014) 33–51 [CrossRef] [MathSciNet] [Google Scholar]
- C. Carstensen and F. Hellwig, Constants in discrete Poincaré and Friedrichs inequalities and discrete quasi-interpolation. CMAM 18, (2017) 433–450 [Google Scholar]
- C. Carstensen and J. Hu, A unifying theory of a posteriori error control for nonconforming finite element methods. Numer. Math. 107, (2007) 473–502 [CrossRef] [MathSciNet] [Google Scholar]
- C. Carstensen and C. Merdon, Computational survey on a posteriori error estimators for nonconforming finite element methods for the Poisson problem. J. Comput. Appl. Math. 249, (2013) 74–94 [Google Scholar]
- C. Carstensen and N. Nataraj, Adaptive Morley FEM for the von Kármán equations with optimal convergence rates. SIAM J. Numer. Anal. 59, (2021) 696–719 [CrossRef] [MathSciNet] [Google Scholar]
- C. Carstensen and N. Nataraj, Mathematics and computation of plates. Under preparation (2021) [Google Scholar]
- C. Carstensen and N. Nataraj, A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified right-hand sides. Comput. Methods Appl. Math. 21, (2021) 289–315 [CrossRef] [MathSciNet] [Google Scholar]
- C. Carstensen and S. Puttkammer, How to prove the discrete reliability for nonconforming finite element methods. J. Comput. Math. 38, (2020) 142–175 [Google Scholar]
- C. Carstensen and S. Puttkammer, Direct guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian. Preprint https://arxiv.org/abs/2105.01505 (2021) [Google Scholar]
- C. Carstensen and H. Rabus, Axioms of adaptivity with separate marking for data resolution. SIAM J. Numer. Anal. 55, (2017) 2644–2665 [CrossRef] [MathSciNet] [Google Scholar]
- 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) 509–558 [CrossRef] [MathSciNet] [Google Scholar]
- 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) 337–353 [CrossRef] [MathSciNet] [Google Scholar]
- 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) 2167–2181 [Google Scholar]
- C. Carstensen, D. Gallistl and N. Nataraj, Comparison results of nonstandard P2 finite element methods for the biharmonic problem. ESAIM: M2AN 49, (2015) 977–990 [CrossRef] [EDP Sciences] [Google Scholar]
- C. Carstensen, G. Mallik and N. Nataraj, A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations. IMA J. Numer. Anal. 39, (2019) 167–200 [Google Scholar]
- C. Carstensen, N. Nataraj, C.R. Gopikrishnan and S. Devika, Unifying a priori and a posteriori error analysis for the lowest-order FEMs in fourth-order semi-linear problems with trilinear nonlinearity. Under preparation, (2021) [Google Scholar]
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland (1978) [Google Scholar]
- G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei and R.L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, (2002) 3669–3750 [Google Scholar]
- X. Feng and O.A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition. Math. Comp. 76, (2007) 1093–1117 (electronic) [CrossRef] [MathSciNet] [Google Scholar]
- D. Gallistl, Morley finite element method for the eigenvalues of the biharmonic operator. IMA J. Numer. Anal. 35, (2015) 1779–1811 [CrossRef] [MathSciNet] [Google Scholar]
- E.H. Georgoulis and P. Houston, Discontinuous Galerkin methods for the biharmonic problem. IMA J. Numer. Anal. 29, (2009) 573–594 [CrossRef] [MathSciNet] [Google Scholar]
- E.H. Georgoulis, P. Houston and J. Virtanen, An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems. IMA J. Numer. Anal. 31, (2011) 281–298 [CrossRef] [MathSciNet] [Google Scholar]
- D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Berlin: Springer-Verlag (2001). Reprint of the 1998 edition [Google Scholar]
- P. Grisvard, Singularities in Boundary Value Problems, Vol. RMA 22. Masson& Springer-Verlag (1992) [Google Scholar]
- T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comp. 79, (2010) 2169–2189 [CrossRef] [MathSciNet] [Google Scholar]
- J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I. Springer-Verlag, New York-Heidelberg (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. (1972), 181 [Google Scholar]
- I. Mozolevski and E. Süli, A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3, (2003) 596–607 [CrossRef] [MathSciNet] [Google Scholar]
- I. Mozolevski, E. Süli and P.R. Bösing, hp-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30, (2007) 465–491 [CrossRef] [MathSciNet] [Google Scholar]
- J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson et Cie, Éditeurs: Paris; Academia, Éditeurs, Prague (1967) [Google Scholar]
- E. Süli and I. Mozolevski, hp-version interior penalty DGFEMs for the biharmonic equation. Comput. Methods Appl. Mech. Eng. 196, (2007) 1851–1863 [Google Scholar]
- L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces. Berlin; Heidelberg: Springer (2010) [Google Scholar]
- A. Veeser and P. Zanotti, Quasi-optimal nonconforming methods for symmetric elliptic problems. I - Abstract theory. SIAM J. Numer. Anal. 56, (2018) 1621–1642 [CrossRef] [MathSciNet] [Google Scholar]
- 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) 2871–2894 [CrossRef] [MathSciNet] [Google Scholar]
- 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) 266–292 [CrossRef] [MathSciNet] [Google Scholar]
- R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford: Numerical Mathematics and Scientific Computation. Oxford University Press (2013) [CrossRef] [Google Scholar]
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