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Volume 16, Issue 1
Analysis of a Fully Discrete Finite Element Method for the Maxwell–Schrödinger System in the Coulomb Gauge

Chupeng Ma, Liqun Cao & Jizu Huang

Int. J. Numer. Anal. Mod., 16 (2019), pp. 139-166.

Published online: 2018-10

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  • Abstract

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell–Schrödinger system in the Coulomb gauge. We propose a fully discrete finite element scheme for the system and prove the conservation of energy and the stability estimates of the scheme. By approximating the vector potential A and the scalar potential $ϕ$ respectively in two finite element spaces satisfying certain orthogonality relation, we tackle the mixed derivative term in the discrete system and make the numerical computations and the theoretical analysis more easier. The existence and uniqueness of solutions to the discrete system are also investigated. The (almost) unconditionally error estimates are derived for the numerical scheme without certain restriction like $τ$ ≤ $Ch$$α$ on the time step $τ$ by using a new technique. Finally, numerical experiments are carried out to support our theoretical analysis.

  • Keywords

Maxwell–Schrödinger, finite element method, energy conserving, error estimates.

  • AMS Subject Headings

65L20, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

machupeng@lsec.cc.ac.cn (Chupeng Ma)

clq@lsec.cc.ac.cn (Liqun Cao)

huangjz@lsec.cc.ac.cn (Jizu Huang)

  • BibTex
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@Article{IJNAM-16-139, author = {Ma , ChupengCao , Liqun and Huang , Jizu}, title = {Analysis of a Fully Discrete Finite Element Method for the Maxwell–Schrödinger System in the Coulomb Gauge}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {16}, number = {1}, pages = {139--166}, abstract = {

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell–Schrödinger system in the Coulomb gauge. We propose a fully discrete finite element scheme for the system and prove the conservation of energy and the stability estimates of the scheme. By approximating the vector potential A and the scalar potential $ϕ$ respectively in two finite element spaces satisfying certain orthogonality relation, we tackle the mixed derivative term in the discrete system and make the numerical computations and the theoretical analysis more easier. The existence and uniqueness of solutions to the discrete system are also investigated. The (almost) unconditionally error estimates are derived for the numerical scheme without certain restriction like $τ$ ≤ $Ch$$α$ on the time step $τ$ by using a new technique. Finally, numerical experiments are carried out to support our theoretical analysis.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12797.html} }
TY - JOUR T1 - Analysis of a Fully Discrete Finite Element Method for the Maxwell–Schrödinger System in the Coulomb Gauge AU - Ma , Chupeng AU - Cao , Liqun AU - Huang , Jizu JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 139 EP - 166 PY - 2018 DA - 2018/10 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12797.html KW - Maxwell–Schrödinger, finite element method, energy conserving, error estimates. AB -

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell–Schrödinger system in the Coulomb gauge. We propose a fully discrete finite element scheme for the system and prove the conservation of energy and the stability estimates of the scheme. By approximating the vector potential A and the scalar potential $ϕ$ respectively in two finite element spaces satisfying certain orthogonality relation, we tackle the mixed derivative term in the discrete system and make the numerical computations and the theoretical analysis more easier. The existence and uniqueness of solutions to the discrete system are also investigated. The (almost) unconditionally error estimates are derived for the numerical scheme without certain restriction like $τ$ ≤ $Ch$$α$ on the time step $τ$ by using a new technique. Finally, numerical experiments are carried out to support our theoretical analysis.

Ma , ChupengCao , Liqun and Huang , Jizu. (2018). Analysis of a Fully Discrete Finite Element Method for the Maxwell–Schrödinger System in the Coulomb Gauge. International Journal of Numerical Analysis and Modeling. 16 (1). 139-166. doi:
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