no. 6
A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2479-2504.

We consider an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016074
Classification : 35J35, 65N30, 65N50
Mots clés : Biharmonic equation, two-energies principle, interior penalty discontinuous Galerkin method, a posteriori error estimator, equilibration
Braess, Dietrich 1 ; Hoppe, R.H.W. 1 ; Linsenmann, Christopher 1

1 
@article{M2AN_2018__52_6_2479_0,
     author = {Braess, Dietrich and Hoppe, R.H.W. and Linsenmann, Christopher},
     title = {A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous {Galerkin} approximation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2479--2504},
     publisher = {EDP-Sciences},
     volume = {52},
     number = {6},
     year = {2018},
     doi = {10.1051/m2an/2016074},
     zbl = {1419.31004},
     mrnumber = {3911627},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016074/}
}
TY  - JOUR
AU  - Braess, Dietrich
AU  - Hoppe, R.H.W.
AU  - Linsenmann, Christopher
TI  - A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2018
SP  - 2479
EP  - 2504
VL  - 52
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2016074/
DO  - 10.1051/m2an/2016074
LA  - en
ID  - M2AN_2018__52_6_2479_0
ER  - 
%0 Journal Article
%A Braess, Dietrich
%A Hoppe, R.H.W.
%A Linsenmann, Christopher
%T A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2018
%P 2479-2504
%V 52
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2016074/
%R 10.1051/m2an/2016074
%G en
%F M2AN_2018__52_6_2479_0
Braess, Dietrich; Hoppe, R.H.W.; Linsenmann, Christopher. A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2479-2504. doi : 10.1051/m2an/2016074. http://www.numdam.org/articles/10.1051/m2an/2016074/

[1] M. Ainsworth and J.T. Oden, Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000). | DOI | MR | Zbl

[2] M. Ainsworth and R. Rankin, Fully computable error bounds for discontinuous Galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes. SIAM J. Numer. Anal. 47 (2010) 4112–4141. | DOI | MR | Zbl

[3] M. Ainsworth and R. Rankin, Constant free error bounds for non-uniform order discontinuous Galerkin finite element approximation on locally refined meshes with hanging nodes. IMA J. Numer. Anal. 31 (2011) 254–280. | DOI | MR | Zbl

[4] J.H. Argyris, I. Fried and D.W. Scharpf, The TUBA family of plate elements for the matrix displacement method. Aero. J. Roy. Aero. Soc. 72 (1968) 701–709.

[5] D. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. ESAIM: M2AN 19 (1985) 7–32. | DOI | Numdam | MR | Zbl

[6] D. Arnold, F. Brezzi, B. Cockburn and D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. | DOI | MR | Zbl

[7] I. Babuška and T. Strouboulis, The Finite Element Method and its Reliability. Clarendon Press, Oxford (2001). | MR

[8] M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22 (2003) 751–756. | DOI | MR | Zbl

[9] E.M. Behrens and J. Guzman, A mixed method for the biharmonic problem based on a system of first order equations. SIAM J. Numer. Anal. 49 (2011) 789–817. | DOI | MR | Zbl

[10] L. Beirao Da Veiga, J. Niiranen and R. Stenberg, A posteriori error estimates for the Morley plate bending element. Numer. Math. 106 (2007) 165–179. | DOI | MR | Zbl

[11] L. Beirao Da Veiga, J. Niiranen and R. Stenberg, A family of C0 finite elements for Kirchhoff plates I: Error analysis. Siam J. Numer. Anal. 45 (2007) 2047–2071. | DOI | MR | Zbl

[12] L. Beirao Da Veiga, J. Niiranen and R. Stenberg, A family of C0 finite elements for Kirchhoff plates II: Numerical results. Comput. Meths. Appl. Mech Engrg. 197 (2008) 1850–1864. | DOI | MR | Zbl

[13] L. Beirao Da Veiga, J. Niiranen and R. Stenberg, A posteriori error analysis for the Morley plate element with general boundary conditions. Int. J. Numer. Meth. Engrg. 83 (2010) 1–26. | DOI | MR | Zbl

[14] D. Braess, Finite Elements, Theory, Fast Solvers and Applications in Solid Mechanics. 3rd ed.. Cambridge University Press, Cambridge (2007). | MR | Zbl

[15] D. Braess, T. Fraunholz and R.H.W. Hoppe, An equilibrated a posteriori error estimator for the interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 52 (2014) 2121–2136. | DOI | MR | Zbl

[16] D. Braess, R.H.W. Hoppe and J. Schöberl, A posteriori estimators for obstacle problems by the hypercircle method. Comput. Visual. Sci. 11 (2008) 351–362. | DOI | MR | Zbl

[17] D. Braess, V. Pillwein and J. Schöberl, Equilibrated residual error estimates are p-robust. Comput. Meth. Appl. Mech. Engrg. 198 (2009) 1189–1197. | DOI | MR | Zbl

[18] D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements. Math. Comput. 77 (2008) 651–672. | DOI | MR | Zbl

[19] S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods. 3rd Edition. Springer, New York (2008). | MR | Zbl

[20] S.C. Brenner, T. Gudi and L.-Y. Sung, An a posteriori error estimator for a quadratic C0-interior penalty method for the biharmonic problem. IMA J. Numer. Anal. 30 (2010) 777–798. | DOI | MR | Zbl

[21] S.C. Brenner and L.-Y. Sung, C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23 (2005) 83–118. | DOI | MR | Zbl

[22] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, Berlin-Heidelberg-New York (1991). | DOI | MR | Zbl

[23] P.G. Ciarlet and P.-A. Raviart, A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, edited by C. De Boor. Academic Press, New York (1974) 125–145. | DOI | MR | Zbl

[24] S. Cochez-Dhondt and S. Nicaise, Equilibrated error estimators for discontinuous Galerkin methods. Numer. Methods Part. Differ. Eq. 24 (2008) 1236–1252. | DOI | MR | Zbl

[25] B. Cockburn, B. Dong and J. Guzman, A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40 (2009) 141–187. | DOI | MR | Zbl

[26] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. | DOI | MR | Zbl

[27] J. Douglas Jr., T. Dupont, P. Percell and R. Scott, A family of C# finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér. 13 (1979) 227–255. | DOI | Numdam | MR | Zbl

[28] 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. Engrg. 191 (2002) 3669–3750. | DOI | MR | Zbl

[29] A. Ern, S. Nicaise and M. Vohralk, An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems. C. R. Acad. Sci. Paris Ser. I 345 (2007) 709–712. | DOI | MR | Zbl

[30] A. Ern and A.F. Stephansen, A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods. J. Comput. Math. 26 (2008) 488–510. | MR | Zbl

[31] A. Ern, A.F. Stephansen and M. Vohralk, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. J. Comput. Appl. Math. 234 (2010) 114–130. | DOI | MR | Zbl

[32] A. Ern and M. Vohralk, Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids. C.R. Acad. Sci. Paris Ser. I 347 (2009) 4441–4444. | MR | Zbl

[33] T. Fraunholz, R.H.W. Hoppe and M. Peter, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the biharmonic problem. J. Numer. Math. 23 (2015) 311–330. | DOI | MR | Zbl

[34] E.H. Georgoulis, P. Houston, Discontinuous Galerkin methods for the biharmonic problem. IMA J. Numer. Anal. 29 (2009) 73–594. | DOI | MR | Zbl

[35] 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. | DOI | MR | Zbl

[36] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985). | MR | Zbl

[37] T. Gudi, N. Nataraj and A.K. Pani, Mixed discontinuous Galerkin finite element method for the biharmonic equation, J. Sci. Comput.. 37 (2008) 139–161. | DOI | MR | Zbl

[38] K. Hellan, Analysis of elastic plates in flexure by a simplified finite element method. Civil Engineering Series 46 (1967). | Zbl

[39] L. Herrmann, Finite element bending analysis for plates. J. Eng. Mech. Div. A.S.C.E. EM5 93 (1967) 13–26.

[40] R.H.W. Hoppe, G. Kanschat and T. Warburton, Convergence analysis of an adaptive interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 47 (2009) 534–550. | DOI | MR | Zbl

[41] X. Huang and J. Huang, A reduced local C0 discontinuous Galerkin method for Kirchhoff plates. Numer. Meth. Part. Diff. Equ. 30 (2014) 1902–1930. | DOI | MR | Zbl

[42] C. Johnson, On the convergence of a mixed finite element method for plate bending problems. Numer. Math. 21 (1973) 43–62. | DOI | MR | Zbl

[43] 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) 1–12. | DOI | MR | Zbl

[44] 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. | DOI | MR | Zbl

[45] P. Neittaanmäki and S. Repin, A posteriori error estimates for boundary-value problems related to the biharmonic equation. East-West J. Numer. Math. 9 (2001) 157–178. | MR | Zbl

[46] L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rat. Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl

[47] W. Prager and J.L. Synge, Approximations in elasticity based on the concept of function spaces. Quart. Appl. Math. 5 (1947) 241–269. | DOI | MR | Zbl

[48] E. Süli and I. Mozolevski, hp-version interior penalty DGFEMs for the biharmonic equation. Comput. Methods Appl. Mech. Eng. 196 (2007) 1851–1863. | DOI | MR | Zbl

[49] J.L. Synge, The method of the hypercircle in function-space for boundary-value problems. Proc. Royal Soc. London, Ser. A 191 (1947) 447–467. | MR | Zbl

[50] J.L. Synge, The Hypercircle Method in Mathematical Physics: A Method for the Approximate Solution of Boundary Value Problems. Cambridge University Press, New York (1957). | MR | Zbl

[51] L. Tartar, Introduction to Sobolev Spaces and Interpolation Theory. Springer, Berlin–Heidelberg–New York (2007).

[52] T. Vejchodsky, Local a posteriori error estimator based on the hypercircle methof. In Proceedings ECCOMAS 2004. edited by P. Neittaanmäki.University of Jyväskylä, Jyväskylä (2004) 16.

[53] G.N. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys. 218 (2006) 860–877. | DOI | MR | Zbl

Cité par Sources :