This work presents a family of stable finite element methods for two- and three-dimensional linear elasticity models. The weak form posed on the skeleton of the partition is a byproduct of the primal hybridization of the elasticity problem. The unknowns are the piecewise rigid body modes and the Lagrange multipliers used to relax the continuity of displacements. They characterize the exact displacement through a direct sum of rigid body modes and solutions to local elasticity problems with Neumann boundary conditions driven by the multipliers. The local problems define basis functions which are in a one-to-one correspondence with the basis of the subspace of Lagrange multipliers used to discretize the problem. Under the assumption that such a basis is available exactly, we prove that the underlying method is well posed, and the stress and the displacement are super-convergent in natural norms driven by (high-order) interpolating multipliers. Also, a local post-processing computation yields strongly symmetric stress which is in local equilibrium and possesses continuous traction on faces. A face-based a posteriori estimator is shown to be locally efficient and reliable with respect to the natural norms of the error. Next, we propose a second level of discretization to approximate the basis functions. A two-level numerical analysis establishes sufficient conditions under which the well-posedness and super-convergent properties of the one-level method is preserved.
DOI : 10.1051/m2an/2015046
Mots clés : Elasticity equation, mixed method, hybrid method, finite element, multiscale, Elasticity equation, mixed method, hybrid method, finite element, multiscale
@article{M2AN_2016__50_2_311_0, author = {Harder, Christopher and Madureira, Alexandre L. and Valentin, Fr\'ed\'eric}, title = {A hybrid-mixed method for elasticity}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {311--336}, publisher = {EDP-Sciences}, volume = {50}, number = {2}, year = {2016}, doi = {10.1051/m2an/2015046}, mrnumber = {3482545}, zbl = {1381.74192}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015046/} }
TY - JOUR AU - Harder, Christopher AU - Madureira, Alexandre L. AU - Valentin, Frédéric TI - A hybrid-mixed method for elasticity JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 311 EP - 336 VL - 50 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015046/ DO - 10.1051/m2an/2015046 LA - en ID - M2AN_2016__50_2_311_0 ER -
%0 Journal Article %A Harder, Christopher %A Madureira, Alexandre L. %A Valentin, Frédéric %T A hybrid-mixed method for elasticity %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 311-336 %V 50 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015046/ %R 10.1051/m2an/2015046 %G en %F M2AN_2016__50_2_311_0
Harder, Christopher; Madureira, Alexandre L.; Valentin, Frédéric. A hybrid-mixed method for elasticity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 311-336. doi : 10.1051/m2an/2015046. http://www.numdam.org/articles/10.1051/m2an/2015046/
A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. 25 (2005) 515–521. | DOI | MR | Zbl
and ,Equilibrium finite elements for the linear elastic problem. Numer. Math. 33 (1979) 367–383. | DOI | MR | Zbl
and ,Multiscale hybrid-mixed method. SIAM J. Numer. Anal. 51 (2013) 3505–3531. | DOI | MR | Zbl
, , and ,Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44 (2006) 1150–1171. | DOI | MR | Zbl
and ,Finite elements for symmetric tensors in three dimensions. Math. Comput. 77 (2008) 1229–1251. | DOI | MR | Zbl
, and ,Mixed finite elements for elasticity. Numer. Math. 92 (2002) 401–419. | DOI | MR | Zbl
and ,Peers: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. | DOI | MR | Zbl
, and ,A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45 (1984) 1–22. | DOI | MR | Zbl
, and ,Nonconforming tetrahedral mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 23 (2014) 783–796. | DOI | MR | Zbl
, and ,Generalized finite element methods: Their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. | DOI | MR | Zbl
and ,Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl
, and ,A locking-free hp dpg method for linear elasticity with symmetric stresses. Numer. Math. 122 (2012) 671–707. | DOI | MR | Zbl
, , and ,Korn’s inequalities for piecewise vector fields. Math. Comput. 73 (2004) 1067–1087. | DOI | MR | Zbl
,S.C. Brenner and L.R. Scott, The Mathematical Foundations of the Finite Element Methods. Springer (2002). | MR
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Vol. 15 of Springer Ser. Comput. Math. Springer-Verlag, Berlin, New-York (1991). | MR | Zbl
A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72 (2002) 541–576. | DOI | MR | Zbl
and ,Superconvergent hdg methods for linear elasticity with weakly symmetric stresses. IMA J. Numer. Anal. (2012) 1–24. | MR | Zbl
and ,A primal dpg method without a first order reformulation. Comput. Math. Appl. 66 (2013) 1058–1064. | DOI | MR
and ,A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer-Verlag, Berlin, New-York (2004). | MR | Zbl
Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal. 49 (2011) 1504–1520. | DOI | MR | Zbl
and ,An analysis of the practical dpg method. Math. Comput. 83 (2014) 537–552. | DOI | MR | Zbl
and ,Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions. Numer. Math. 126 (2014) 153–171. | DOI | MR | Zbl
and ,A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys. 245 (2013) 107–130. | DOI | MR | Zbl
, and ,On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogenous coefficients. SIAM Multiscale Model. Simul. 13 (2015) 491–518. | DOI | MR | Zbl
, and ,A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. | DOI | MR | Zbl
and ,J. Hu, A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation. Preprint [math.NA] (2015). | arXiv | MR
An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl
and ,Basis of finite element methods for solid continua, Int. J. Numer. Methods Engrg. 1 (1969) 3–28. | DOI | Zbl
and ,W. Qiu and K. Shi, An hdg method for linear elasticity with strong symmetric stresses. Preprint [math.NA] (2014). | arXiv
P. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspect of finite element methods, No. 606 in Lect. Notes Math. Springer-Verlag, New York (1977) 292–315. | MR | Zbl
Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comput. 31 (1977) 391–413. | DOI | MR | Zbl
and ,A hybridizable discontinuous galerkin method for linear elasticity. Int. J. Numer. Methods Engrg. 80 (2009) 1058–1092. | DOI | MR | Zbl
, and ,On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48 (1986) 447–462. | DOI | MR | Zbl
,a. Toselli and O. Widlund, Domain decomposition methods-algorithms and theory. Vol. 34 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (2005). | MR | Zbl
Cité par Sources :