计算数学
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计算数学  2017, Vol. 39 Issue (1): 42-58    DOI:
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一类随机非自伴波方程的半离散有限元近似
李晓翠1, 杨小远1, 张英晗2
1. 北京航空航天大学数学与系统科学学院, 北京 100191;
2. 北京科技大学数理学院, 北京 100083
SEMIDISCRETE FINITE ELEMENT APPROXIMATION OF STOCHASTIC NONSELFADJOINT WAVE EQUATION
Li Xiaocui1, Yang Xiaoyuan1, Zhang Yinghan2
1. Department of Mathematics, Beihang University, LMIB of the Ministry of Education, Beijing 100191, China;
2. School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
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摘要 本文研究了由白噪音驱动的随机非自伴波方程的有限元近似,由于线性算子A非自伴,不能应用A的特征值和特征向量,从而得到的结果更具有一般性.空间离散上采用标准的有限元法,并借助强连续算子函数的性质,得到了该方程的强收敛误差估计.本文方法也适用于多维情况的分析.最后用数值算例验证了理论分析的正确性.
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关键词随机非自伴波方程   有限元方法   非自伴算子   cosine算子函数   强收敛     
Abstract: We study the semidiscrete finite element approximation of the linear stochastic nonselfadjoint wave equation forced by additive noise.The results here are more general since the linear operator A does not need to be self-adjoint and we do not need information about eigenvalues and eigenfunctions of the linear operator A.In order to obtain the strong convergence error estimates,a standard finite element method for the spatial discretisation and the properties of a strongly continuous operator cosine function are used.The error estimates are applicable in the multi-dimensional case.
Key wordsstochastic nonselfadjoint wave equation   finite element method   nonselfadjoint operator   cosine operator function   strong convergence   
收稿日期: 2015-12-30;
基金资助:

国家自然科学基金(61271010),北京市自然科学基金(4152029)资助项目.

引用本文:   
. 一类随机非自伴波方程的半离散有限元近似[J]. 计算数学, 2017, 39(1): 42-58.
. SEMIDISCRETE FINITE ELEMENT APPROXIMATION OF STOCHASTIC NONSELFADJOINT WAVE EQUATION[J]. Mathematica Numerica Sinica, 2017, 39(1): 42-58.
 
[1] Jin B, Lazarov R, Pasciak J and Zhou Z. Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H-s, 0< s ≤ 1. Lecture Notes in Computer Science, 2013, 24-37. -s, 0 target="_blank">
[2] Travis C and Webb G. Cosine families and abstract nonlinear second order differential equations[J]. Acta Mathematica Academiae Scientiaum Hungaricae, 1978, 32:75-96.
[3] Prévôt C and Röckner R. A concise course on stochastic partial differential equations. Berlin:2007, vol. 1905 of Lecture Notes in Mathematics, Springer.
[4] Cohen D and Sigg M. Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations[J]. Numerische Mathematik, 2012, 121:1-29.
[5] Cohen D, Larsson S and Sigg M. A trigonometric method for the linear stochastic wave equation[J]. SIAM J. Numer. Anal., 2013, 51:204-222.
[6] Lutz D. On bounded time-dependent perturbations of operator cosine functions. Aequationes Mathematicae, 1981, 23:197-203.
[7] Allen E, Novosel S and Zhang Z. Finite element and difference approximation of some linear stochastic partial differential equations[J]. Stoch. Stoch. Rep., 1998, 64(1-2):117-142.
[8] Baker G and Bramble J. Semidiscrete and single step full discrete approximations for second order hyperbolic equations[J]. RAIRO Numer. Anal., 1979, 13:75-100.
[9] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 1992.
[10] Lord G and Tambue A. Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise[J]. IMA J. Numer. Anal., 2012, 1-29.
[11] Fujita H and Suzuki T. Evolutions problems (part1). in:P. G. Ciarlet, J. L. Lions (Eds.) Handbook of Numerical Analysis, vol II, North-Holland, Amsterdam, 1991, 789-928.
[12] J. Printems. On the discretization in time of parabolic stochastic partial differential equations. Esaim Mathematical Modelling and Numerical Analysis, 2001, 1055-1078.
[13] Quer-Sardanyons L and Sanz-Solé M. Space semi-discretisations for a stochastic wave equation[J]. Potential Anal., 2006, 24:303-332.
[14] Palla M, Sofi A and Muscolino G. Nonlinear random vibrations of a suspended cable under wind loading. Proceedings of Fourth International Conference on Computational Stochstic Mechanics(CSM4), 2002, 159-168.
[15] Pazy A. Semigroups of linear operators and applications to partial differential equations. SpringerVerlag, Berlin, 1983.
[16] Kovács M, Larsson S and Lindgren F. Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise[J]. BIT Number.Math., 2012, 52(1):85-108.
[17] Walsh J B. On numerical solutions of the stochastic wave equation[J]. Illinois J. Math., 2006, 50:991-1018.
[18] Kovács M, Larsson S and Saedpanah F. Finite element approximation of the linear stochastic wave equation with additive noise[J]. SIAM J. Numer. Anal., 2010, 48(2):408-427.
[19] Ciarlet P. The finite element method for elliptic problems. North-Holland:1978.
[20] Du Q and Zhang T. Numerical approximation of some linear stochastic partial differential equations driven by special additive noise[J]. SIAM J. Numer. Anal., 2002, 40:1421-1445.
[21] Anton R, Cohen D, Larsson S and Wang X. Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise[J]. SIAM J. Numer. Anal., arXiv:1503.00073v1.
[22] Dalang R, Khoshnevisan D, Mueller C, Nualart D and Xiao Y. A minicourse on stochastic partial differential equation. Berlin:2009, vol.1962 of Lecture Notes in Mathematics, Springer-Verlag.
[23] Larsson S. Nonsmooth data error estimates with applications to the study of longtime behavior of finite element solutions of semilinear parabolic problems. 1992.
[24] Georgios T, Georgios E. Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise. ESAIM:Mathematical Modelling and Numerical Analysis, 2010, 44:289-322.
[25] Thomée V. Galerkin finite element methods for parabolic problems. 2006, volume 25 of Springer Series in Computational Mathematics, Springer-Verlag:Berlin.
[26] Wang X. An exponential integrator scheme for time discretization of nonlinear stochastic wave equation[J]. Journal of Scientific Computing, 2014, 64:234-263.
[27] Yang X, Li X, Qi R and Zhang Y. Full-discrete finite element method for stochastic hyperbolic equation[J]. Journal of Computational Mathematics, 2015, 33(5):533-556.
[28] Yan Y. Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise[J]. BIT Numer. Math., 2004, 44:829-847.
[29] Yan Y. Galerkin finite element methods for stochastic parabolic partial differential equations[J]. SIAM J. Numer. Anal., 2005, 43(4):1363-1384.
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