# Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schr\"{o}dinger equations under the Lorentz gauge

## TL;DR

This paper develops and analyzes an alternating Crank-Nicolson finite element method for solving the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge, providing optimal error estimates and confirming results through numerical experiments.

## Contribution

It introduces a new numerical scheme with proven optimal error bounds for Maxwell-Schrödinger equations under the Lorentz gauge.

## Key findings

- Optimal error estimates are established for the proposed method.
- Numerical experiments confirm the theoretical convergence rates.

## Abstract

In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schr\"{o}dinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is presented and the optimal error estimate for the numerical algorithm is obtained by a mathematical inductive method. Numerical examples are then carried out to confirm the theoretical results.