# Mathematical and numerical analysis of the time-dependent Maxwell--Schr\"{o}dinger Equations in the Coulomb gauge

## TL;DR

This paper establishes the global existence of weak solutions for the time-dependent Maxwell--Schrödinger equations in the Coulomb gauge, proposes an energy-conserving finite element scheme, and confirms its accuracy through numerical tests.

## Contribution

It introduces a novel energy-conserving finite element method for the Maxwell--Schrödinger system and provides rigorous error analysis without time-step restrictions.

## Key findings

- Proved global existence of weak solutions.
- Developed an energy-conserving finite element scheme.
- Achieved optimal error estimates and validated with numerical results.

## Abstract

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schr\"{o}dinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an energy-conserving fully discrete finite element scheme for the system and prove the existence and uniqueness of solutions to the discrete system. The optimal error estimates for the numerical scheme without any time-step restrictions are then derived. Numerical results are provided to support our theoretical analysis.