Finite Element Methods For Wave Propagation With Debye Polarization In Nonlinear Dielectric Materials
This work provides rigorous numerical analysis for a specific nonlinear wave propagation model, which is incremental for researchers in computational electromagnetics.
The paper develops finite element methods for wave propagation with Debye polarization in nonlinear dielectric materials, proving well-posedness and establishing convergence rates of O(Δt) in time and O(Δt+h^s) in space. Numerical examples validate the theoretical results.
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 $O(Δt)$. 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(Δt+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.