# A Crank-Nicolson Finite Element Method and the Optimal Error Estimates for the modified Time-dependent Maxwell-Schr\"{o}dinger Equations

## TL;DR

This paper introduces a Crank-Nicolson finite element method for the time-dependent Maxwell-Schrödinger equations, providing optimal error estimates and validating them through numerical tests, advancing numerical analysis for quantum-electromagnetic interactions.

## Contribution

The paper develops a new finite element scheme with optimal error bounds for Maxwell-Schrödinger equations without time-step restrictions.

## Key findings

- Optimal energy-norm error estimates derived
- Numerical tests confirm theoretical error bounds
- Method applicable to quantum-electromagnetic interaction problems

## Abstract

In this paper we consider the initial-boundary value problem for the time-dependent Maxwell-Schr\"{o}dinger equations, which arises in the interaction between the matter and the electromagnetic field for the semiconductor quantum devices. A Crank-Nicolson finite element method for solving the problem is presented. The optimal energy-norm error estimates for the numerical algorithm without any time-step restrictions are derived. Numerical tests are then carried out to confirm the theoretical results.