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随机平面线弹性问题的一类弱Galerkin方法

陈明卿, 谢小平   

  1. 四川大学数学学院, 成都 610064
  • 收稿日期:2019-12-17 出版日期:2021-08-15 发布日期:2021-08-20
  • 通讯作者: 谢小平,Email:xpxie@scu.edu.cn.
  • 基金资助:
    国家自然科学基金(11771312).

陈明卿, 谢小平. 随机平面线弹性问题的一类弱Galerkin方法[J]. 计算数学, 2021, 43(3): 279-300.

Chen Mingqing, Xie Xiaoping. WEAK GALERKIN FINITE ELEMENT METHODS FOR STOCHASTIC LINEAR PLANE ELASTICITY EQUATIONS[J]. Mathematica Numerica Sinica, 2021, 43(3): 279-300.

WEAK GALERKIN FINITE ELEMENT METHODS FOR STOCHASTIC LINEAR PLANE ELASTICITY EQUATIONS

Chen Mingqing, Xie Xiaoping   

  1. School of Mathematics, Sichuan University, Chengdu 610064, China
  • Received:2019-12-17 Online:2021-08-15 Published:2021-08-20
本文针对带有随机杨氏模量和荷载的平面线弹性问题,提出了一类随机弱Galerkin有限元方法.先利用Karhunen-Loève展开把随机项参数化,将方程转化为一个确定性问题;再采用弱Galerkin有限元法和$k$-/$p$-型方法分别离散空间区域和随机场.在弱Galerkin离散中,用分片$s(s\geqslant 1$)和$s+1$次多项式逼近单元内部的应力和位移,用分片$s$次多项式逼近位移在单元边界上的迹.证明了该方法关于空间网格尺度最优且与Lamé常数$\lambda$一致无关的误差估计.最后通过数值算例验证了理论结果.
This paper proposes a class of stochastic weak Galerkin finite element methods for solving linear plane elasticity problems with stochastic Young's modulus and loads. Firstly, we convert the original system to a deterministic one by Karhuen-Loève expansion for Parameterizing the stochastic terms. Then we use a weak Galerkin (WG) discretization in the spatial domain and a $k$-/$p$- version method in the stochastic field. The WG method adopts piecewise-polynomial approximations of degrees $s(s\geqslant 1)$ and $s+1$ for the stress and displacement respectively, and $s$ for the displacement trace on the element boundaries. Optimal error estimates are derived which are uniform with respect to the Lamé constant $\lambda$. Finally, numerical experiments are performed to verify the theoretical results.

MR(2010)主题分类: 

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[1] Babuška I, Chatzipantelidis P. On solving elliptic stochastic partial differential equations[J]. Comput. Methods. Appl. M., 2010, 191(37):4093-4122.
[2] Babuška I, Nobile F, Tempone R. A stochastic collocation method for elliptic partial differential equations with random input data[J]. SIAM J. Numer. Anal.,2007, 45(3):1005-1034.
[3] Babuška I, Tempone R, Zouraris G E. Galerkin finite element approximations of stochastic elliptic partial differential equations[J]. SIAM J. Numer. Anal., 2005, 42(2):800-825.
[4] Brenner S C, Scott L R. The Mathematical Theory of Finite Element Methods[M]. 233 Spring Street, New York, NY 10013:Springer Science & Business Media LLC, 2008.
[5] Chen G, Feng M, Xie X. Robust globally divergence-free weak Galerkin methods for Stokes equations[J]. J. Comput. Math., 2016, 34(5):549-572.
[6] Chen G, Xie X. A robust weak galerkin finite element method for linear elasticity with strong symmetric stresses[J]. Comput. Methods Appl. Math., 2016, 16(3):389-408.
[7] Chen L, Wang J, Wang Y, Ye X. An auxiliary space multigrid preconditioner for the weak Galerkin method[J]. Comput. Math. Appl., 2015, 70(4):330-344.
[8] Chen L, Wang J, Ye X. A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems[J]. J. Sci. Comput., 2014, 59(2):496-511.
[9] Chen Y, Chen G, Xie X. Weak Galerkin finite element method for Biot s consolidation problem[J]. J. Comput. Appl. Math., 2018, 330:398-416.
[10] Deb M K, Babuška I M, Oden J T. Solution of stochastic partial differential equations using galerkin finite element techniques[J]. Comput.Methods.Appl.Mech Engrg., 2001, 190(48):6359-6372.
[11] Frauenfelder P, Schwab C, Todor R A. Finite elements for elliptic problems with stochastic coefficients[J]. Comput. Methods Appl. Mech. Engrg., 2005, 194(2-5):205-228.
[12] Ghanem R G, Spanos P D. Polynomial chaos in stochastic finite elements[J]. J. Appl. Mech., 1990, 57(1):197.
[13] Ghanem R G, Spanos P D. Stochastic finite elements:A spectral approach[M]. Springer-Verlag, 1992.
[14] Han Y, Li H, Xie X. Robust globally divergence-free weak Galerkin finite element methods for unsteady natural convection problems[J]. Numer. Math-Theory Me., 2019, 12(4):1266-1308.
[15] Han Y, Xie X. Robust globally divergence-free weak Galerkin finite element methods for? natural convection problems[J]. Commun. Comput. Phys., 2019, 26:1039-1070
[16] Hu X, Mu L, Ye X, Weak Galerkin method for the Biot's consolidation model[J]. Comput. Math. Appl., 2018, 75(6):2017-2030.
[17] Bramble J H, Lazarov R D, Pasciak J E. Least-Squares Methods For Linear Elasticity Based On A Discrete Minus One Inner Product[J]. Comput. Method Appl. M., 2001, 191(8-10):727-744.
[18] Kamiński M. Generalized perturbation-based stochastic finite element method in elastostatics[J]. Comput. Struct., 2007, 85:586-594.
[19] Li B, Xie X. A two-level algorithm for the weak Galerkin discretization of diffusion problems[J]. J. Comput. Appl. Math., 2015, 287:179-195.
[20] Li B, Xie X. BPX preconditioner for nonstandard finite element methods for diffusion problems[J]. SIAM J. Numer. Anal., 2016, 54(2):1147-1168.
[21] Li B, Xie X, Zhang S. BPS preconditioners for a weak Galerkin finite element method for 2D diffusion problems with strongly discontinuous coefficients[J]. Comput. Math. Appl., 2018.
[22] Liu X, Li J, Chen Z. A weak Galerkin finite element method for the Navier-Stokes equations[J]. J. Comput. Appl. Math., 2018,333:442-457.
[23] Lucor D, Su C H, Karniadakis G E. Generalized polynomial chaos and random oscillators[J]. Int. J. Numer. Methods Engrg., 2004, 60(3):571-596.
[24] Mu L, Wang J, Ye X, Zhang S. A weak Galerkin finite element method for the Maxwell equations[J]. J. Sci. Comput., 2015, 65(1):363-386.
[25] Narayan A, Xiu D. Stochastic collocation methods on unstructured grids in high dimensions via interpolation[J]. SIAM. J. Sci. Comput., 2012, 34(3):1729-1752.
[26] Nobile F, Tempone R, Webster C G. A sparse grid stochastic collocation method for partial differential equations with random input data[J]. SIAM J. Numer. Anal., 2008, 46(5):2309-2345.
[27] Nobile F, Tempone R, Webster C G. An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data[J]. SIAM J. Numer. Anal., 2008, 46(5):2411-2442.
[28] Oksendal B. Stochastic differential equations:an introduction with applications, 5th ed[M]. Berlin:Springer-Verlag,1998.
[29] Riesz F, Sznagy B. Functional analysis[M]. Dover Publications, 1990.
[30] Schwab C, Todor R A. Karhuen-loève approximation of random fields by generalized fast multipole methods[J]. J. Compu. Phys., 2006, 217:100-122.
[31] Sharif R. A Galerkin isogeometric method for Karhunen-Loève approximation of random fields[J]. Comput. Methods Appl. Mech. Engrg., 2018, 338:533-561.
[32] Spanos P D, Ghanem R. Stochastic finite element expansion for random media[J]. J. Eng. MechAsce., 1989, 115(5):1035-1053.
[33] Wang C, Wang J, Wang R, Zhang R. A locking-free weak Galerkin finite element method for elasticity problems in the primal formulation[J]. J. Comput. Appl. Math., 2016, 307:346-366.
[34] Wang C, Wang J. An effcient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes[J]. Adv. Comput., 2013, 68(12):2314-2330.
[35] Wang J, Ye X. A weak galerkin finite element method for second order elliptic problems[J]. J. Comput. Appl. Math., 2013, 241:103-115.
[36] Wang J, Ye X. A weak galerkin mixed finite element method for second order elliptic problems[J]. Math. Comp., 2014, 83(289):2101-2126.
[37] Wang J, Ye X. A weak Galerkin finite element method for the Stokes equations[J]. Adv. Comput. Math., 2016, 42(1):155-174.
[38] 王军平, 叶秀, 张然.弱有限元方法简论.计算数学, 2016, 38(3):289-308.
[39] Wang R, Wang X, Zhai Q, Zhang R. A weak Galerkin finite element scheme for solving the stationary Stokes equations[J]. J. Comput. Appl. Math., 2016, 302:171-185.
[40] Wiener N. The homogeneous chaos[J]. Am. J. Mathe., 1938, 60(4):897-936.
[41] Xie H, Zhai Q, Zhang R, Zhang Z. The weak Galerkin method for elliptic eigenvalue problems[J]. Commun. Comput. Phy., 2019, 26(1):160-191.
[42] Xiu D. Fast numerical methods for stochastic computations:A review[J]. Commun. Comput.Phys., 2009, 5(2-4):242-272.
[43] Xiu D, Karniadakis G E. The wiener-askey polynomial chaos for stochastic differential equations[J]. SIAM. J. Sci. Comput., 2002, 24(2):619-644.
[44] Xiu D, Karniadakis G E. Modeling uncertainty in flow simulations via generalized polynomial chaos[J]. J. Comput. Phys., 2003, 187(1):137-167.
[45] Xu X, Fan W, Xie X. Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations[J]. Int. J. Numer. Meth. Engng., 2016.
[46] Zhai Q, Zhang R, Wang X. A hybridized weak Galerkin finite element scheme for the Stokes equations[J]. Sci. China. Math., 2015, 58(11):2455-2472.
[47] Zhang J, Zhang K, Li J, Wang X. A Weak Galerkin Finite Element Method for the Navier-Stokes Equations[J]. Commun. Comput. Phys., 2018, 23:706-746.
[48] Zhang R, Zhai Q. A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order[J]. J. Sci. Comput., 2015, 64(2):559-585.
[49] Zhang T, Lin T. A posteriori error estimate for a modified weak Galerkin method solving elliptic problems[J]. Numer. Methods Partial Differential Eq., 2017, 33:381-398.
[50] Zhang T, Lin T. A stable weak galerkin finite element method for stokes problem[J]. J. Comput. Appl. Math., 2018, 333:235-246.
[51] Zhang T, Lin T. An analysis of a weak Galerkin finite element method for stationary Navier-Stokes problems[J]. J. Comput. Appl. Math., 2018.
[52] Zheng X, Chen G, Xie X. A divergence-free weak Galerkin method for quasi-Newtonian Stokes flows[J]. Sci. China. Math., 2017, 60(8):1039-1070.
[53] Zheng X, Xie X. A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem[J]. East Asian Journal on Applied Mathematics,, 2017, 7(3):508-529.
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