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Volume 29, Issue 2
Finite Element Method with Superconvergence for Nonlinear Hamiltonian Systems

Chuanmiao Chen, Qiong Tang & Shufang Hu

DOI: 10.4208/jcm.1009-m3108

J. Comp. Math., 29 (2011), pp. 167-184.

Published online: 2011-04

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  • Abstract

This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects: conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes $t_n$ for mid-long time and demonstrate the symplecticity of high accuracy. The proofs depend strongly on superconvergence analysis. Numerical experiments show that the proposed method can preserve the energy very well and also can make the global trajectory error small for a long time.

  • Keywords

Nonlinear Hamiltonian systems, Finite element method, Superconvergence, Energy conservation, Symplecticity, Trajectory.

  • AMS Subject Headings

65N30.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-29-167, author = {Chuanmiao Chen, Qiong Tang and Shufang Hu}, title = {Finite Element Method with Superconvergence for Nonlinear Hamiltonian Systems}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {2}, pages = {167--184}, abstract = {

This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects: conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes $t_n$ for mid-long time and demonstrate the symplecticity of high accuracy. The proofs depend strongly on superconvergence analysis. Numerical experiments show that the proposed method can preserve the energy very well and also can make the global trajectory error small for a long time.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1009-m3108}, url = {http://global-sci.org/intro/article_detail/jcm/8471.html} }
TY - JOUR T1 - Finite Element Method with Superconvergence for Nonlinear Hamiltonian Systems AU - Chuanmiao Chen, Qiong Tang & Shufang Hu JO - Journal of Computational Mathematics VL - 2 SP - 167 EP - 184 PY - 2011 DA - 2011/04 SN - 29 DO - http://doi.org/10.4208/jcm.1009-m3108 UR - https://global-sci.org/intro/article_detail/jcm/8471.html KW - Nonlinear Hamiltonian systems, Finite element method, Superconvergence, Energy conservation, Symplecticity, Trajectory. AB -

This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects: conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes $t_n$ for mid-long time and demonstrate the symplecticity of high accuracy. The proofs depend strongly on superconvergence analysis. Numerical experiments show that the proposed method can preserve the energy very well and also can make the global trajectory error small for a long time.

Chuanmiao Chen, Qiong Tang and Shufang Hu. (2011). Finite Element Method with Superconvergence for Nonlinear Hamiltonian Systems. Journal of Computational Mathematics. 29 (2). 167-184. doi:10.4208/jcm.1009-m3108
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