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A new dual-mixed finite element method for the plane linear elasticity problem with pure traction boundary conditions. (English) Zbl 1169.74601

Summary: We consider the stress-displacement-rotation formulation of the plane linear elasticity problem with pure traction boundary conditions and develop a new dual-mixed finite element method for approximating its solution. The main novelty of our approach lies on the weak enforcement of the non-homogeneous Neumann boundary condition through the introduction of the boundary trace of the displacement as a Lagrange multiplier. A suitable combination of PEERS and continuous piecewise linear functions on the boundary are employed to define the dual-mixed finite element scheme. We apply the classical Babuška-Brezzi theory to show the well-posedness of the continuous and discrete formulations. Then, we derive a priori rates of convergence of the method, including an estimate for the global error when the stresses are measured with the \(L^{2}\)-norm. Finally, several numerical results illustrating the good performance of the mixed finite element scheme are reported.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
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[1] Arnold, D.N.; Brezzi, F.; Douglas, J., PEERS: A new mixed finite element method for plane elasticity, Jpn. J. appl. math., 1, 347-367, (1984) · Zbl 0633.73074
[2] Arnold, D.N.; Douglas, J.; Gupta, Ch.P., A family of higher order mixed finite element methods for plane elasticity, Numer. math., 45, 1-22, (1984) · Zbl 0558.73066
[3] Arnold, D.N.; Falk, R.S.; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta numer., 15, 1-155, (2006) · Zbl 1185.65204
[4] Arnold, D.N.; Falk, R.S.; Winther, R., Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. comput., 76, 260, 1699-1723, (2007) · Zbl 1118.74046
[5] Arnold, D.N.; Winther, R., Mixed finite elements for elasticity, Numer. math., 92, 401-419, (2002) · Zbl 1090.74051
[6] Babuška, I., The finite element method with Lagrangian multipliers, Numer. math., 20, 179-192, (1973) · Zbl 0258.65108
[7] Babuška, I.; Aziz, A.K., Survey lectures on the mathematical foundations of the finite element method, () · Zbl 0268.65052
[8] Babuška, I.; Gatica, G.N., On the mixed finite element method with Lagrange multipliers, Numer. methods partial differ. eq., 19, 2, 192-210, (2003) · Zbl 1021.65056
[9] Braess, D., Finite elements, Theory, fast solvers, and applications in solid mechanics, (1997), Cambridge University Press · Zbl 0894.65054
[10] Brenner, S.C., A nonconforming mixed multigrid method for the pure traction problem in planar linear elasticity, Math. comput., 63, 208, 435-460, (1994) · Zbl 0809.73064
[11] Brenner, S.C.; Scott, L.R., The mathematical theory of finite element methods, (1994), Springer-Verlag, Inc. New York · Zbl 0804.65101
[12] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer Verlag · Zbl 0788.73002
[13] Falk, R.S., Nonconforming finite element methods for the equations of linear elasticity, Math. comput., 57, 529-550, (1991) · Zbl 0747.73044
[14] Gatica, G.N., Analysis of a new augmented mixed finite element method for linear elasticity allowing \(\mathbb{RT}_0 - \mathbb{P}_1 - \mathbb{P}_0\) approximations, Math. modell. numer. anal. (ESAIM), 40, 1, 1-28, (2006)
[15] Gatica, G.N.; Márquez, A.; Meddahi, S., Analysis of the coupling of primal and dual-mixed finite element methods for a two-dimensional fluid – solid interaction problem, SIAM J. numer. anal., 45, 5, 2072-2097, (2007) · Zbl 1225.74087
[16] Gatica, G.N.; Wendland, W.L., Coupling of mixed finite elements and boundary elements for a hyperelastic interface problem, SIAM J. numer. anal., 34, 6, 2335-2356, (1997) · Zbl 0895.73067
[17] Grisvard, P., Elliptic problems in non-smooth domains, Monographs and studies in mathematics, 24, (1985), Pitman · Zbl 0695.35060
[18] Grisvard, P., Probléms aux limites dans LES polygones. mode démploi, Bull. direct. etudes recherches (serie C), 1, 21-59, (1986) · Zbl 0623.35031
[19] Hiptmair, R., Finite elements in computational electromagnetism, Acta numer., 11, 237-339, (2002) · Zbl 1123.78320
[20] Johnson, C.; Mercier, B., Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. math., 30, 1, 103-116, (1978) · Zbl 0427.73072
[21] Lee, C.-O., A conforming mixed finite element method for the pure traction problem of linear elasticity, Appl. math. comput., 93, 1, 11-29, (1998) · Zbl 0969.74579
[22] Lonsing, M.; Verfürth, R., On the stability of BDMS and PEERS elements, Numer. math., 99, 1, 131-140, (2004) · Zbl 1076.65090
[23] Prössdorf, S.; Silbermann, B., Numerical analysis for integral and related operator equations, (1991), Birkhäuser-Verlag Basel · Zbl 0763.65103
[24] Roberts, J.E.; Thomas, J.M., Mixed and hybrid methods, () · Zbl 0875.65090
[25] Stenberg, R., A family of mixed finite elements for the elasticity problem, Numer. math., 53, 5, 513-538, (1988) · Zbl 0632.73063
[26] Yi, S.-Y., A new non-conforming mixed finite element method for linear elasticity, Math. models methods appl. sci., 16, 7, 979-999, (2006) · Zbl 1094.74057
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