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 - <p style="text-align: justify;">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 <strong>A</strong> 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 $τ$ ≤ $Ch$<sup>$α$</sup>&nbsp;on the time step $τ$ by using a new technique. Finally, numerical experiments are carried out to support our theoretical analysis.</p>