2021/09/17 by Tomáš Dohnal, Dohnal, Tomáš, Giulio Romani +3
Computer Science · Engineering · Mathematics · #35J62 35R05 35B65 35B45 35Q61 78M45 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2109.08513
openalex publication_date 2021/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Maxwell equations in the absence of free charges require initial data with a divergence free displacement field \mathcal D. In materials in which the dependence \mathcal D=\mathcal D(\mathcal E) is nonlinear the quasilinear problem ∇⋅\mathcal D(\mathcal E)=0 is hence to be solved. In many applications, e.g. in the modelling of wave-packets, an approximative asymptotic ansatz of the electric field \mathcal E is used, which satisfies this divergence condition at t=0 only up to a small residual. We search then for a small correction of the ansatz to enforce ∇⋅\mathcal D(\mathcal E)=0 at t=0 and choose this correction in the form of a gradient field. In the usual case of a power type nonlinearity in \mathcal D(\mathcal E) this leads to the sum of the Laplace and p-Laplace operators. We also allow for the medium to consist of two different materials so that a transmission problem across an interface is produced. We prove the existence of the correction term for a general class of nonlinearities and provide regularity estimates for its derivatives, independent of the L2-norm of the original ansatz. In this way, when applied to the wave-packet setting, the correction term is indeed asymptotically smaller than the original ansatz. We also provide numerical experiments to support our analysis.