Splitting positive definite mixed element method for viscoelasticity wave equation

doi:10.1007/s11464-012-0183-8

Front Math Chin

2012, Vol. 7

Issue (4) : 725-742 https://doi.org/10.1007/s11464-012-0183-8

RESEARCH ARTICLE

Splitting positive definite mixed element method for viscoelasticity wave equation

Yang LIU¹(

), Hong LI¹(

), Wei GAO¹, Siriguleng HE¹, Jinfeng WANG²

1. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China; 2. School of Statistics and Mathematics, Inner Mongolia Finance and Economics College, Hohhot 010051, China

Download: PDF(463 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method.

Keywords Viscoelasticity wave equation transformation splitting positive definite system mixed finite element method error estimate

Corresponding Author(s): LIU Yang,Email:mathliuyang@yahoo.cn; LI Hong,Email:smslh@imu.edu.cn

Issue Date: 01 August 2012

Cite this article:

Yang LIU,Hong LI,Wei GAO, et al. Splitting positive definite mixed element method for viscoelasticity wave equation[J]. Front Math Chin, 2012, 7(4): 725-742.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-012-0183-8
https://academic.hep.com.cn/fmc/EN/Y2012/V7/I4/725

1	Adams R A. Sobolev Spaces. New York: Academic, 1975
2	Brezzi F, Douglas J Jr, Fortin M, Marini L D. Efficient rectangular mixed finite elements in two and three space variables. RAIRO Modèl Math Anal Numér , 1987, 21: 581-604
3	Brezzi F, Douglas J Jr, Marini L D. Two families of mixed finite elements for second order elliptic problems. Numer Math , 1985, 47: 217-235 doi: 10.1007/BF01389710
4	Chen Y P, Huang Y Q. The superconvergence of mixed finite element methods for nonlinear hyperbolic equations. Commun Nonlinear Sci Numer Simul , 1998, 3(3): 155-158 doi: 10.1016/S1007-5704(98)90006-5
5	Chen Z X. Finite Element Methods and Their Applications. Berlin: Springer-Verlag, 2005
6	Chen Z X. Implementation of mixed methods as finite difference methods and applications to nonisothermal multiphase flow in porous media. J Comput Math , 2006, 24(3): 281-294
7	Ciarlet P G. The Finite Element Methods for Elliptic Problems. New York: North-Holland, 1978
8	Douglas J Jr, Ewing R, Wheeler M F. The approximation of the pressure by a mixed method in the simulation of miscible displacement. RARIO Anal Numer , 1983, 17: 17-33
9	Ewing R E, Lin Y P, Wang J P, Zhang S H. L^∞-error estimates and superconvergence in maximum norm of mixed finite element methods for nonfickian flows in porous media. Internat J Numer Anal Model , 2005, 2(3): 301-328
10	Gao L P, Liang D, Zhang B. Error estimates for mixed finite element approximations of the viscoelasticity wave equation. Math Methods Appl Sci , 2004, 27: 1997-2016 doi: 10.1002/mma.534
11	Guo H, Rui H X. Least-squares Galerkin procedures for pseudo-hyperbolic equations. Appl Math Comput , 2007, 189: 425-439 doi: 10.1016/j.amc.2006.11.094
12	Jiang Z W, Chen H Z. Errors estimates for mixed finite element methods for Sobolev equation. Northeast Math J , 2001, 17(3): 301-314
13	Johnson C, Thomée V. Error estimates for some mixed finite element methods for parabolic problems. RARIO Anal Numer , 1981, 15: 41-78
14	Li H R, Luo Z D, Li Q. Generalized difference methods and numerical simulation for two-dimensional viscoelastic problems. Math Numer Sin , 2007, 29(3): 257-262 (in Chinese)
15	Li J C. Multiblock mixed finite element methods for singularly perturbed problems. Appl Numer Math , 2000, 35: 157-175 doi: 10.1016/S0168-9274(99)00055-0
16	Liu Y, Li H. H¹-Galerkin mixed finite element methods for pseudo-hyperbolic equations. Appl Math Comput , 2009, 212: 446-457 doi: 10.1016/j.amc.2009.02.039
17	Liu Y, Li H, He S. Mixed time discontinuous space-time finite element method for convection diffusion equations. Appl Math Mech , 2008, 29(12): 1579-1586 doi: 10.1007/s10483-008-1206-y
18	Luo Z D. Theory Bases and Applications of Finite Element Mixed Methods. Beijing: Science Press, 2006 (in Chinese)
19	Pani A K, Yuan J Y. Mixed finite element methods for a strongly damped wave equation. Numer Methods Partial Differential Equations , 2001, 17: 105-119 doi: 10.1002/1098-2426(200103)17:2<105::AID-NUM2>3.0.CO;2-F
20	Raviart P A, Thomas J M. A mixed finite element methods for second order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Lecture Notes in Math, Vol 606 . Berlin: Springer, 1977, 292-315
21	Shi Y H, Shi D Y. Superconvergence analysis and extrapolation of ACM finite element methods for viscoelasticity equation. Math Appl , 2009, 22(3): 534-541
22	Yang D P. A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media. Numer Methods Partial Differential Equations , 2001, 17: 229-249 doi: 10.1002/num.3
23	Zhang J S, Yang D P. A splitting positive definite mixed element method for secondorder hyperbolic equations. Numer Methods Partial Differential Equations , 2009, 25: 622-636 doi: 10.1002/num.20363

[1]	Yuhan ZHANG, Junyang GAO, Jianyong QIAO, Qinghua WANG. Dynamics of a family of rational maps concerning renormalization transformation[J]. Front. Math. China, 2020, 15(4): 807-833.
[2]	Danda ZHANG, Da-jun ZHANG. Addition formulae, Backlund transformations, periodic solutions, and quadrilateral equations[J]. Front. Math. China, 2019, 14(1): 203-223.
[3]	Ruishu WANG, Xiaoshen WANG, Kai ZHANG, Qian ZHOU. Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form[J]. Front. Math. China, 2018, 13(5): 1121-1140.
[4]	Liang WEI, Zhiping LI. Fourier-Chebyshev spectral method for cavitation computation in nonlinear elasticity[J]. Front. Math. China, 2018, 13(1): 203-226.
[5]	Jun HU, Jing ZHANG, Yabo WU. Involutions in Weyl group of type F4[J]. Front. Math. China, 2017, 12(4): 891-906.
[6]	Hongfei FU,Hongxing RUI,Zhaojie ZHOU. A posteriori error estimates for optimal control problems constrained by convection-diffusion equations[J]. Front. Math. China, 2016, 11(1): 55-75.
[7]	Zhi LI,Jiaowan LUO. Transportation inequalities for stochastic delay evolution equations driven by fractional Brownian motion[J]. Front. Math. China, 2015, 10(2): 303-321.
[8]	Wei ZHANG,Weidong ZHAO. Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis[J]. Front. Math. China, 2015, 10(2): 415-434.
[9]	Xiumei XING,Lei JIAO. Boundedness of semilinear Duffing equations with singularity[J]. Front. Math. China, 2014, 9(6): 1427-1452.
[10]	Miao WANG,Jiang-Lun WU. Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations[J]. Front. Math. China, 2014, 9(3): 601-622.
[11]	Haiqiong ZHAO, Zuonong ZHU. A semidiscrete Gardner equation[J]. Front Math Chin, 2013, 8(5): 1099-1115.
[12]	Christian SCIMITERNA, Decio LEVI. Classification of discrete equations linearizable by point transformation on a square lattice[J]. Front Math Chin, 2013, 8(5): 1067-1076.
[13]	Yingnan ZHANG, Yi HE, Hon-Wah TAM. One variant of a (2+ 1)-dimensional Volterra system and its (1+ 1)-dimensional reduction[J]. Front Math Chin, 2013, 8(5): 1085-1097.
[14]	Pengzhan HUANG, Yinnian HE, Xinlong FENG. Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations[J]. Front Math Chin, 2013, 8(4): 837-854.
[15]	Siriguleng HE, Hong LI, Yang LIU. Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations[J]. Front Math Chin, 2013, 8(4): 825-836.

Viewed

Full text

Abstract

Cited

Shared

Discussed