Wavenumber-Explicit hp-FEM Analysis for Maxwell's Equations with Impedance Boundary Conditions.

Area of Science:

• Computational electromagnetics
• Mathematical physics
• Numerical analysis

Background:

• Solving time-harmonic Maxwell equations is crucial in electromagnetics.
• High wavenumber regimes pose significant challenges for numerical simulations.
• Analytic boundary conditions and impedance boundary conditions are common in physical applications.

Purpose of the Study:

• To develop a wavenumber-explicit stability and regularity theory for high wavenumber Maxwell equations.
• To analyze the quasi-optimality of Galerkin discretization using Nédélec elements.
• To establish conditions for accurate numerical solutions in challenging electromagnetic scenarios.

Main Methods:

• Decomposition of the solution into finite Sobolev regularity and analytic parts.
• Development of a wavenumber-explicit stability and regularity theory.
• Analysis of Galerkin discretization with Nédélec elements of order p on a mesh with size h.

Main Results:

• A wavenumber-explicit theory is established, controlling solution components uniformly in k.
• Quasi-optimality of the Galerkin discretization is demonstrated under specific scale resolution conditions.
• The analysis provides a foundation for understanding numerical solution behavior in high wavenumber regimes.

Conclusions:

• The developed theory ensures stability and regularity for high wavenumber problems.
• Galerkin discretization with Nédélec elements is shown to be quasi-optimal under defined conditions.
• This work advances the numerical solution of electromagnetic problems with analytic boundaries and impedance conditions.