Primal stabilized hybrid and DG finite element methods for the linear elasticity problem. (English) Zbl 1362.74005
Summary: Primal stabilized hybrid finite element methods for the linear elasticity problem are proposed consisting of locally discontinuous Galerkin problems in the primal variable coupled to a global problem in the multiplier which is identified with the trace of the displacement field. Numerical analysis, covering both continuous or discontinuous interpolations of the multiplier, shows that the proposed formulation preserves the main properties of the associate DG method such as consistency, stability, boundedness and optimal rates of convergence in the energy norm, and in the \(\mathbf L^2(\varOmega)\) norm for adjoint consistent formulations. Convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here and a local post-processing technique is proposed to recover stress approximations with improved rates of convergence in \(\mathbf H(\operatorname{div})\) norm.
MSC:
| 74B05 | Classical linear elasticity |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
linear elasticity; discontinuous Galerkin; hybridization; stabilization; local post-processingReferences:
| [1] | Babuška, I., Error bounds for finite element method, Numer. Math., 16, 322-333 (1970-1971) · Zbl 0214.42001 |
| [2] | Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, Rev. Fr. Automat. Inform., 8-R2, 129-151 (1974) · Zbl 0338.90047 |
| [3] | Arnold, D. N.; Winther, R., Mixed finite elements for elasticity, Numer. Math., 92, 3, 401-419 (2002) · Zbl 1090.74051 |
| [4] | Arnold, D. N.; Winther, R., Nonconforming mixed elements for elasticity, Math. Models Methods Appl. Sci., 13, 3, 295-307 (2003) · Zbl 1057.74036 |
| [5] | Arnold, D. N.; Awanou, G., Rectangular mixed finite elements for elasticity, Math. Models Methods Appl. Sci., 15, 9, 1417-1429 (2005) · Zbl 1077.74044 |
| [6] | Chen, S.-C.; Yang, Y.-N., Conforming rectangular mixed finite elements for elasticity, J. Sci. Comput., 47, 1, 93-108 (2011) · Zbl 1248.74038 |
| [7] | Awanou, G., Two remarks on rectangular mixed finite elements for elasticity, J. Sci. Comput., 50, 1, 91-102 (2012) · Zbl 1314.74058 |
| [8] | Adams, S.; Cockburn, B., A mixed finite element method for elasticity in three dimensions, J. Sci. Comput., 25, 3, 515-521 (2005) · Zbl 1125.74382 |
| [9] | Arnold, D.; Awanou, G., The serendipity family of finite elements, Found. Comput. Math., 11, 3, 337-344 (2011) · Zbl 1218.65125 |
| [10] | Arnold, D. N.; Brezzi, F.; Douglas, J., PEERS: a new mixed finite element for plane elasticity, Japan J. Appl. Math., 1, 2, 347-367 (1984) · Zbl 0633.73074 |
| [11] | Arnold, D. N.; Falk, R. S.; Winther, R., Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76, 260, 1699-1723 (2007) · Zbl 1118.74046 |
| [12] | Qiu, W.; Demkowicz, L., Mixed hp-finite element method for linear elasticity with weakly imposed symmetry, Comput. Methods Appl. Mech. Engrg., 198, 47-48, 3682-3701 (2009) · Zbl 1230.74195 |
| [13] | Awanou, G., Rectangular mixed elements for elasticity with weakly imposed symmetry condition, Adv. Comput. Math., 38, 2, 351-367 (2013) · Zbl 1333.74079 |
| [14] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag · Zbl 0788.73002 |
| [15] | Hughes, T. J.R.; Franca, L. P., A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all higher-order spaces, Comput. Methods Appl. Mech. Engrg., 67, 2, 223-240 (1988) · Zbl 0611.73077 |
| [16] | Franca, L. P.; Hughes, T. J.R.; Loula, A. F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation, Numer. Math., 53, 1-2, 123-141 (1988) · Zbl 0656.73036 |
| [17] | Franca, L. P.; Stenberg, R., Error analysis of some Galerkin least squares methods for the elasticity equations, SIAM J. Numer. Anal., 28, 6, 1680-1697 (1991) · Zbl 0759.73055 |
| [18] | Arnold, D. N.; Falk, R. S., A uniformly accurate finite element method for the Reissner-Mindlin plate, SIAM J. Numer. Anal., 26, 6, 1276-1290 (1989) · Zbl 0696.73040 |
| [19] | Kouhia, R.; Stenberg, R., A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Engrg., 124, 3, 195-212 (1995) · Zbl 1067.74578 |
| [20] | Brenner, S. C.; Sung, L.-Y., Linear finite element methods for planar linear elasticity, Math. Comp., 200, 321-338 (1992) · Zbl 0766.73060 |
| [21] | Lew, A.; Neff, P.; Sulsky, D.; Ortiz, M., Optimal BV estimates for a discontinuous Galerkin method for linear elasticity, Appl. Math. Res. Express, 2004, 3, 73-106 (2004) · Zbl 1115.74021 |
| [22] | Rivière, B.; Wheeler, M. F., Optimal error estimates for discontiuous Galerkin methods applied to linear elasticity problems, TICAM Report, 1-10 (2000) |
| [23] | Wihler, T. P., Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems, Math. Comp., 75, 255, 1087-1102 (2006) · Zbl 1088.74047 |
| [24] | Hansbo, P.; Larson, M. G., Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Engrg., 191, 17-18, 1895-1908 (2002) · Zbl 1098.74693 |
| [25] | Hansbo, P.; Larson, M. G., Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, ESAIM Math. Model. Numer. Anal., 37, 1, 63-72 (2003) · Zbl 1137.65431 |
| [26] | Chen, Y.; Huang, J.; Huang, X.; Xu, Y., On the local discontinuous Galerkin method for linear elasticity, Math. Probl. Eng., 2010, 1-20 (2010) · Zbl 1426.74284 |
| [27] | Cockburn, B.; Schötzau, D.; Wang, J., Discontinuous Galerkin methods for incompressible elastic materials, Comput. Methods Appl. Mech. Engrg., 195, 25-28, 3184-3204 (2006) · Zbl 1128.74041 |
| [28] | Bustinza, R., A note on the local discontinuous Galerkin method for linear problems in elasticity, Sci. Ser. A Math. Sci., 13, 72-83 (2006) · Zbl 1344.74056 |
| [29] | Ewing, R. E.; Wang, J.; Yang, Y., A stabilized discontinuous finite element method for elliptic problems, Numer. Linear Algebra Appl., 10, 1-2, 83-104 (2003) · Zbl 1071.65551 |
| [30] | Cockburn, B.; Gopalakrishnan, J.; Sayas, F.-J., A projection-based error analysis of HDG methods, Math. Comp., 79, 271, 1351-1367 (2010) · Zbl 1197.65173 |
| [31] | Cockburn, B.; Dong, B.; Guzmán, J., A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems, Math. Comp., 77, 264, 1887-1916 (2008) · Zbl 1198.65193 |
| [32] | Cockburn, B.; Dong, B.; Guzmán, J.; Restelli, M.; Sacco, R., A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems, SIAM J. Sci. Comput., 31, 5, 3827-3846 (2009) · Zbl 1200.65093 |
| [33] | Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312 |
| [34] | Nguyen, N. C.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228, 23, 8841-8855 (2009) · Zbl 1177.65150 |
| [35] | Cockburn, B.; Gopalakrishnan, J.; Nguyen, N. C.; Peraire, J.; Sayas, F.-J., Analysis of HDG methods for Stokes flow, Math. Comp., 80, 274, 723-760 (2011) · Zbl 1410.76164 |
| [36] | Hughes, T. J.R.; Scovazzi, G.; Bochev, P. B.; Buffa, A., A multiscale discontinuous Galerkin method with the computational structure of a continuous Galerkin method, Comput. Methods Appl. Mech. Engrg., 195, 19-22, 2761-2787 (2006) · Zbl 1124.76027 |
| [37] | Buffa, A.; Hughes, T. J.R.; Sangalli, G., Analysis of a multiscale discontinuous Galerkin method for convection-diffusion problems, SIAM J. Numer. Anal., 44, 4, 1420-1440 (2006) · Zbl 1153.76038 |
| [38] | Arruda, N. C.B.; Loula, A. F.D.; Almeida, R. C., Locally discontinuous but globally continuous Galerkin methods for elliptic problems, Comput. Methods Appl. Mech. Engrg., 255, 104-120 (2013) · Zbl 1297.65143 |
| [39] | Raviart, P. A.; Thomas, J. M., Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp., 31, 138, 391-413 (1977) · Zbl 0364.65082 |
| [41] | Labeur, R. J.; Wells, G. N., A Galerkin interface stabilisation method for the advection-diffusion and incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 196, 49-52, 4985-5000 (2007) · Zbl 1173.76344 |
| [42] | Wells, G. N., Analysis of an interface stabilized finite element method: the advection-diffusion-reaction equation, SIAM J. Numer. Anal., 49, 1-2, 87-109 (2011) · Zbl 1226.65097 |
| [43] | Adams, R. A., (Sobolev Spaces. Sobolev Spaces, Pure and Applied Mathematics (1975), Academic Press: Academic Press New York) · Zbl 0314.46030 |
| [44] | Lai, W. M.; Rubin, D.; Krempl, E., Introduction to Continuum Mechanics (1999), Butterworth Heinemann: Butterworth Heinemann Woburn, MA |
| [45] | Gurtin, M. E., An Introduction to Continuum Mechanics (1981), Academic Press: Academic Press New York, NY · Zbl 0559.73001 |
| [46] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080 |
| [47] | Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19, 4, 742-760 (1982) · Zbl 0482.65060 |
| [48] | Brenner, S. C., Korn’s inequalities for piecewise H1 vector fields, Math. Comp., 73, 247, 1067-1087 (2004) · Zbl 1055.65118 |
| [49] | Rivière, B., Discontinuous Galerkin Methods For Solving Elliptic and Parabolic Equations: Theory and Implementation (2008), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 1153.65112 |
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