Frontiers of Mathematics in China >
Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form
Received date: 22 Nov 2017
Accepted date: 19 Sep 2018
Published date: 29 Oct 2018
Copyright
A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.
Ruishu WANG , Xiaoshen WANG , Kai ZHANG , Qian ZHOU . Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form[J]. Frontiers of Mathematics in China, 2018 , 13(5) : 1121 -1140 . DOI: 10.1007/s11464-018-0730-z
| 1 |
Adams S, Cockburn B. A mixed finite element method for elasticity in three dimensions. J Sci Comput, 2005, 25: 515–521
|
| 2 |
Arnold D N,Brezzi F. Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél Math Anal Numér, 1985, 19: 7–32
|
| 3 |
Arnold D N, Brezzi F, Douglas J. PEERS: a new mixed finite element for plane elasticity. Japan J Appl Math, 1984, 1: 347–367
|
| 4 |
Arnold D N, Douglas J, Gupta C P. A family of higher order mixed finite element methods for plane elasticity. Numer Math, 1984, 45: 1–22
|
| 5 |
Arnold D N, Falk R S, Winther R. Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math Comp, 2007, 76: 1699–1723
|
| 6 |
Arnold D N, Winther R. Mixed finite elements for elasticity. Numer Math, 2002, 92: 401–419
|
| 7 |
Baiocchi C, Brezzi F, Marini L D. Stabilization of Galerkin methods and applications to domain decomposition. In: Lecture Notes in Comput Sci, Vol 653. Berlin: Springer, 1992, 345–355
|
| 8 |
Brezzi F. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev Francaise Automat Informat Recherche Opérationnelle Sér, 1974, 8: 129–151
|
| 9 |
Brezzi F, Douglas J, Marini L D. Two families of mixed finite elements for second order elliptic problems. Numer Math, 1985, 47: 217–235
|
| 10 |
Chen G,Feng M, Xie X. Robust globally divergence-free weak Galerkin methods for Stokes equations. J Comput Math, 2016, 34: 549–572
|
| 11 |
Chen G, Feng M, Xie X. A class of robust WG/HDG finite element method for convection-diffusion-reaction equations. J Comput Appl Math, 2017, 315: 107–125
|
| 12 |
Chen G, Xie X. A robust weak Galerkin finite element method for linear elasticity with strong symmetric stresses. Comput Methods Appl Math, 2016, 16: 389–408
|
| 13 |
Chen L, Wang J, Ye X. A posteriori error estimates for weak Galerkin finite element methods for second order elliptic problems. J Sci Comput, 2014, 59: 496–511
|
| 14 |
Johnson C, Mercier B. Some equilibrium finite element methods for two-dimensional elasticity problems. Numer Math, 1978, 30: 103–116
|
| 15 |
Li B, Xie X. A two-level algorithm for the weak Galerkin discretization of diffusion problems. J Comput Appl Math, 2015, 287: 179–195
|
| 16 |
Li J, Wang X, Zhang K. Multi-level Monte Carlo weak Galerkin method for elliptic equations with stochastic jump coecients. Appl Math Comput, 2016, 275: 181–194
|
| 17 |
Mu L, Wang J, Ye X. A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J Comput Phys, 2014, 273: 327–342
|
| 18 |
Mu L, Wang J, Ye X. Weak Galerkin finite element methods on polytopal meshes. Int J Numer Anal Model, 2015, 12: 31–53
|
| 19 |
Mu L, Wang J, Ye X. A hybridized formulation for the weak Galerkin mixed finite element method. J Comput Appl Math, 2016, 307: 335–345
|
| 20 |
Mu L, Wang J, Ye X, Zhang S. A C0-weak Galerkin finite element method for the biharmonic equation. J Sci Comput, 2014, 59: 473–495
|
| 21 |
Qiu W, Demkowicz L. Mixed hp-finite element method for linear elasticity with weakly imposed symmetry. Comput Methods Appl Mech Engrg, 2009, 198: 3682–3701
|
| 22 |
Wang C, Wang J, Wang R, Zhang R. A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation. J Comput Appl Math, 2016, 307: 346–366
|
| 23 |
Wang J, Ye X. A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math, 2013, 241: 103–115
|
| 24 |
Wang J, Ye X. A weak Galerkin mixed finite element method for second order elliptic problems. Math Comp, 2014, 83: 2101–2126
|
| 25 |
Wang J, Ye X. A weak Galerkin finite element method for the Stokes equations. Adv Comput Math, 2016, 42: 155–174
|
| 26 |
Wang R, Wang X, Zhai Q, Zhang R. A weak Galerkin finite element scheme for solving the stationary Stokes equations. J Comput Appl Math, 2016, 302: 171–185
|
| 27 |
Wang R, Zhang R. A weak Galerkin finite element method for the linear elasticity problem in mixed form. J Comput Math, 2018, 36: 469–491
|
| 28 |
Watwood V B Jr, Hartz B J. An equilibrium stress field model for finite element solutions of two-dimensional elastostatic problems. Internat J Solids Structures, 1968, 4: 857–873
|
| 29 |
Zhai Q, Zhang R, Mu L. A new weak Galerkin finite element scheme for the Brinkman model. Commun Comput Phys, 2016, 19: 1409{1434
|
| 30 |
Zhang R, Zhai Q. A weak Galerkin finite element scheme for the biharmonic equations by using polynomials of reduced order. J Sci Comput, 2015, 64: 559–585
|
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