Analysis of a Fully Discrete Finite Element Method for the Maxwell–Schrödinger System in the Coulomb Gauge
Year: 2019
Author: Chupeng Ma, Liqun Cao, Jizu Huang
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 1 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2019-IJNAM-12797
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 1 : pp. 139–166
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Maxwell–Schrödinger finite element method energy conserving error estimates.