Finite Element Methods For Wave Propagation With Debye Polarization In Nonlinear Dielectric Materials

TL;DR

This paper develops finite element methods for simulating wave propagation with Debye polarization in nonlinear dielectric materials, providing theoretical analysis and numerical validation of the schemes' accuracy and convergence.

Contribution

It introduces a decoupled full-discrete scheme combining Euler time discretization and Raviart-Thomas-Nédélec elements, with convergence analysis under minimal regularity assumptions.

Findings

Established well-posedness of the model using Rother's method.

Derived error estimates with order O(Δt+h^s).

Validated the theoretical results with numerical examples.

Abstract

In this paper, we consider the wave propagation with Debye polarization in nonlinear dielectric materials. For this model, the Rother's method is employed to derive the well-posedness of the electric fields and the existence of the polarized fields by monotonicity theorem as well as the boundedness of the two fields are established. Then, the time errors are derived for the semi-discrete solutions by the order $O(\Delta t)$. Subsequently, decoupled the full-discrete scheme of the Euler in time and Raviart-Thomas-N$\acute{e}$d$\acute{e}$lec element $k\geq 2$ in spatial is established. Based on the truncated error, we present the convergent analysis with the order $O(\Delta t+h^s) $ under the technique of a-prior $L^\infty$ assumption. For the $k=1$, we employ the superconvergence technique to ensure the a-prior $L^\infty$ assumption. In the end, we give some numerical examples to demonstrate our theories.