# Analysis of a new augmented mixed finite element method for linear elasticity allowing ${\mathrm{\mathbb{R}\mathbb{T}}}_{0}$-${\mathbb{P}}_{1}$-${\mathbb{P}}_{0}$ approximations

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

- Volume: 40, Issue: 1, page 1-28
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topGatica, Gabriel N.. "Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations." ESAIM: Mathematical Modelling and Numerical Analysis 40.1 (2006): 1-28. <http://eudml.org/doc/249755>.

@article{Gatica2006,

abstract = {
We present a new stabilized mixed finite element method for the linear elasticity problem in $\mathbb\{R\}^2$. The
approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and
equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that
the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that
the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions,
respectively. In particular, the discrete scheme allows the utilization of Raviart–Thomas spaces of lowest
order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the
rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which
yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is
then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh
size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the
domain. Several numerical results illustrating the good performance of the augmented mixed finite element
scheme in the case of Dirichlet boundary conditions are also reported.
},

author = {Gatica, Gabriel N.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Mixed-FEM; augmented formulation; linear elasticity; locking-free.; locking-free},

language = {eng},

month = {2},

number = {1},

pages = {1-28},

publisher = {EDP Sciences},

title = {Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb\{RT\}_0$-$\mathbb\{P\}_1$-$\mathbb\{P\}_0$ approximations},

url = {http://eudml.org/doc/249755},

volume = {40},

year = {2006},

}

TY - JOUR

AU - Gatica, Gabriel N.

TI - Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2006/2//

PB - EDP Sciences

VL - 40

IS - 1

SP - 1

EP - 28

AB -
We present a new stabilized mixed finite element method for the linear elasticity problem in $\mathbb{R}^2$. The
approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and
equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that
the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that
the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions,
respectively. In particular, the discrete scheme allows the utilization of Raviart–Thomas spaces of lowest
order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the
rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which
yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is
then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh
size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the
domain. Several numerical results illustrating the good performance of the augmented mixed finite element
scheme in the case of Dirichlet boundary conditions are also reported.

LA - eng

KW - Mixed-FEM; augmented formulation; linear elasticity; locking-free.; locking-free

UR - http://eudml.org/doc/249755

ER -

## References

top- D.N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element method for plane elasticity. Japan J. Appl. Math.1 (1984) 347–367.
- D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element method for the Stokes equations. Calcolo21 (1984) 337–344.
- D.N. Arnold, J. Douglas and Ch.P. Gupta, A family of higher order mixed finite element methods for plane elasticity. Numer. Math.45 (1984) 1–22.
- D. Arnold and R. Falk, Well-posedness of the fundamental boundary value problems for constrained anisotropic elastic materials. Arch. Rational Mech. Analysis98 (1987) 143–190.
- I. Babuška and G.N. Gatica, On the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differential Equations19 (2003) 192–210.
- M. Barrientos, G.N. Gatica and E.P. Stephan, A mixed finite element method for nonlinear elasticity: two-fold saddle point approach and a posteriori error estimate. Numer. Math.91 (2002) 197–222.
- T.P. Barrios, G.N. Gatica and F. Paiva, A wavelet-based stabilization of the mixed finite element method with Lagrange multipliers. Appl. Math. Lett. (in press).
- D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press (1997).
- F. Brezzi and J. Douglas, Stabilized mixed methods for the Stokes problem. Numer. Math.53 (1988) 225–235.
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).
- F. Brezzi and M. Fortin, A minimal stabilisation procedure for mixed finite element methods. Numer. Math.89 (2001) 457–491.
- F. Brezzi, J. Douglas and L.D. Marini, Variable degree mixed methods for second order elliptic problems. Mat. Apl. Comput.4 (1985) 19–34.
- F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.
- F. Brezzi, M. Fortin and L.D. Marini, Mixed finite element methods with continuous stresses. Math. Models Methods Appl. Sci.3 (1993) 275–287.
- D. Chapelle and R. Stenberg, Locking-free mixed stabilized finite element methods for bending-dominated shells, in Plates and shells (Quebec, QC, 1996), American Mathematical Society, Providence, RI, CRM Proceedings Lecture Notes21 (1999) 81–94.
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1978).
- J. Douglas and J. Wan, An absolutely stabilized finite element method for the Stokes problem. Math. Comput.52 (1989) 495–508.
- H.-Y. Duan and G.-P. Liang, Analysis of some stabilized low-order mixed finite element methods for Reissner-Mindlin plates. Comput. Methods Appl. Mech. Engrg.191 (2001) 157–179.
- L.P. Franca, New Mixed Finite Element Methods. Ph.D. Thesis, Stanford University (1987).
- L.P. Franca and T.J.R. Hughes, Two classes of finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1988) 89–129.
- L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg.142 (1997) 361–364.
- L.P. Franca and R. Stenberg, Error analysis of Galerkin least squares methods for the elasticity equations. SIAM J. Numer. Anal.28 (1991) 1680–1697.
- C.O. Horgan, Korn's inequalities and their applications in continuum mechanics. SIAM Rev.37 (1995) 491–511.
- C.O. Horgan and J.K. Knowles, Eigenvalue problems associated with Korn's inequalities. Arch. Rational Mech. Anal.40 (1971) 384–402.
- C.O. Horgan and L.E. Payne, On inequalities of Korn, Friedrichs and Babuška-Aziz. Arch. Rational Mech. Analysis82 (1983) 165–179.
- N. Kechkar and D. Silvester, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comput.58 (1992) 1–10.
- G. Lube and A. Auge, Stabilized mixed finite element approximations of incompressible flow problems. Zeitschrift für Angewandte Mathematik und Mechanik72 (1992) T483–T486.
- W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000).
- A. Masud and T.J.R. Hughes, A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Engrg.191 (2002) 4341–4370.
- R. Nascimbene and P. Venini, A new locking-free equilibrium mixed element for plane elasticity with continuous displacement interpolation. Comput. Methods Appl. Mech. Engrg191 (2002) 1843–1860.
- S. Norburn and D. Silvester, Fourier analysis of stabilized ${Q}_{1}$-${Q}_{1}$ mixed finite element approximation. SIAM J. Numer. Anal.39 (2001) 817–833.
- R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math.53 (1988) 513–538.
- T. Zhou, Stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems. Math. Comput.72 (2003) 1655–1673.
- T. Zhou and L. Zhou, Analysis of locally stabilized mixed finite element methods for the linear elasticity problem. Chinese J. Engrg Math.12 (1995) 1–6.

## Citations in EuDML Documents

top- Tomás P. Barrios, Gabriel N. Gatica, María González, Norbert Heuer, A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity
- Tomás P. Barrios, Gabriel N. Gatica, María González, Norbert Heuer, A residual based error estimator for an augmented mixed finite element method in linear elasticity

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.