Mixed finite element methods for linear elasticity and the Stokes equations based on the Helmholtz decomposition
Schedensack, Mira1
1 Institut für Numerische Simulation, Universität Bonn, Wegelerstraße 6, 53115 Bonn, Germany.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 399-425.

This paper introduces new mixed finite element methods (FEMs) of degree ≥1 for the equations of linear elasticity and the Stokes equations based on Helmholtz decompositions. These FEMs are robust with respect to the incompressible limit and also allow for mixed boundary conditions. Adaptive algorithms driven by efficient and reliable residual-based error estimators are introduced and proved to converge with optimal rate in the case of the Stokes equations with pure Dirichlet boundary.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016024
Classification : 65N30, 76M10, 65N12
Mots-clés : Linear elasticity, Stokes equations, non-conforming FEM, Helmholtz decomposition, mixed FEM, adaptive FEM, optimality
Schedensack, Mira 1

1 Institut für Numerische Simulation, Universität Bonn, Wegelerstraße 6, 53115 Bonn, Germany. 
@article{M2AN_2017__51_2_399_0,
     author = {Schedensack, Mira},
     title = {Mixed finite element methods for linear elasticity and the {Stokes} equations based on the {Helmholtz} decomposition},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {399--425},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/m2an/2016024},
     mrnumber = {3626404},
     zbl = {1398.76125},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2016024/}
}
TY  - JOUR
AU  - Schedensack, Mira
TI  - Mixed finite element methods for linear elasticity and the Stokes equations based on the Helmholtz decomposition
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 399
EP  - 425
VL  - 51
IS  - 2
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2016024/
DO  - 10.1051/m2an/2016024
LA  - en
ID  - M2AN_2017__51_2_399_0
ER  - 
%0 Journal Article
%A Schedensack, Mira
%T Mixed finite element methods for linear elasticity and the Stokes equations based on the Helmholtz decomposition
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 399-425
%V 51
%N 2
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2016024/
%R 10.1051/m2an/2016024
%G en
%F M2AN_2017__51_2_399_0
Schedensack, Mira. Mixed finite element methods for linear elasticity and the Stokes equations based on the Helmholtz decomposition. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 399-425. doi : 10.1051/m2an/2016024. https://www.numdam.org/articles/10.1051/m2an/2016024/

D.N. Arnold, F. Brezzi and J. Douglas, Jr. PEERS: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. | DOI | MR | Zbl

D.N. Arnold and R. Winther, Mixed finite elements for elasticity. Numer. Math. 92 (2002) 401–419. | DOI | MR | Zbl

D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1985) 337–344. | DOI | MR | Zbl

D.N. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions. Math. Comp. 77 (2008) 1229–1251. | DOI | MR | Zbl

E. Bänsch, P. Morin and R.H. Nochetto, An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 40 (2002) 1207–1229. | DOI | MR | Zbl

R. Becker and S. Mao, Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations. SIAM J. Numer. Anal. 49 (2011) 970–991. | DOI | MR | Zbl

P. Binev and R. Devore, Fast computation in adaptive tree approximation. Numer. Math. 97 (2004) 193–217. | DOI | MR | Zbl

P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. | DOI | MR | Zbl

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). | MR | Zbl

S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics, 3rd edition. Springer Verlag, New York, Berlin, Heidelberg (2008). | Zbl

D. Braess, Finite Elements. 3rd edition. Cambridge University Press, Cambridge (2007). | MR | Zbl

S.C. Brenner and L.-Y. Sung, Linear finite element methods for planar linear elasticity. Math. Comp. 59 (1992) 321–338. | DOI | MR | Zbl

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974) 129–151. | Numdam | MR | Zbl

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). | MR | Zbl

C. Carstensen, A unifying theory of a posteriori finite element error control. Numer. Math. 100 (2005) 617–637. | DOI | MR | Zbl

C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) 187–209. | DOI | MR | Zbl

C. Carstensen and H. Rabus, Axioms of adaptivity for separate marking. Preprint (2016). | arXiv | MR

C. Carstensen and M. Schedensack, Medius analysis and comparison results for first-order finite element methods in linear elasticity. IMA J. Numer. Anal. 35 (2014) 1591–1621. | DOI | MR | Zbl

C. Carstensen, D. Gallistl and M. Schedensack, Quasi-optimal adaptive pseudostress approximation of the Stokes equations. SIAM J. Numer. Anal. 51 (2013) 1715–1734. | DOI | MR | Zbl

C. Carstensen, D. Peterseim and H. Rabus, Optimal adaptive nonconforming FEM for the Stokes problem. Numer. Math. 123 (2013) 291–308. | DOI | MR | Zbl

P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Operationnelle 9 (1975) 77–84. | Numdam | MR | Zbl

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) 33–75. | Numdam | MR | Zbl

M. Crouzeix and R.S. Falk, Nonconforming finite elements for the Stokes problem. Math. Comput. 52 (1989) 437–456. | DOI | MR | Zbl

M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles. Internat. J. Numer. Methods Engrg. 19 (1983) 505–520. | DOI | MR | Zbl

T. Gantumur, Optimal convergence rates for an adaptive mixed finite element method for the Stokes problem. Preprint (2014). | arXiv

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). | MR | Zbl

J. Guzmán and M. Neilan, Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83 (2014) 15–36. | DOI | MR | Zbl

J. Hu and J. Xu, Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem. J. Sci. Comput. 55 (2013) 125–148. | DOI | MR | Zbl

R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg. 124 (1995) 195–212. | DOI | MR | Zbl

C. Kreuzer and M. Schedensack, Instance optimal Crouzeix–Raviart adaptive finite element methods for the Poisson and Stokes problems. IMA J. Numer. Anal. 36 (2015) 593–617. | DOI | MR | Zbl

L. Morley, The triangular equilibrium element in the solution of plate bending problems. Aeronaut.Quart. 19 (1968) 149–169. | DOI

H. Rabus, On the quasi-optimal convergence of adaptive nonconforming finite element methods in three examples. Ph. D. thesis, Humboldt-Universität zu Berlin (2014).

P.-A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. In Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975). Springer, Berlin (1977) 292–315. | MR | Zbl

M. Schedensack, A class of mixed finite element methods based on the Helmholtz decomposition in computational mechanics. Ph. D. thesis, Humboldt-Universität zu Berlin (2015).

M. Schedensack, A new discretization for mth-Laplace equations with arbitrary polynomial degrees. SIAM J. Numer. Anal. 54 (2016) 2138–2162. | DOI | MR | Zbl

M. Schedensack, A new generalization of the P1 non-conforming FEM to higher polynomial degrees. Comput. Methods Appl. Math. 17 (2017) 161–185. | DOI | MR | Zbl

L.R. Scott and M. Vogelius, Conforming finite element methods for incompressible and nearly incompressible continua. In Large-scale Computations in Fluid Mechanics, Part 2 (La Jolla, Calif., 1983). Vol. 22 of Lect. Appl. Math.. Amer. Math. Soc. Providence, RI (1985) 221–244. | MR | Zbl

R. Stevenson, The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77 (2008) 227–241. | DOI | MR | Zbl

A. Veeser, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16 (2016) 723–750. | DOI | MR | Zbl

R. Verfürth,A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in numerical mathematics. Wiley (1996). | Zbl

Cité par Sources :