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Semi-discrete finite element approximation applied to Maxwell's equations in nonlinear media

Lutz Angermann

TL;DR

This work analyzes a semi-discrete in space finite element approximation of Maxwell's equations in Kerr-type nonlinear media, using Nédélec elements of the first family. It establishes energy stability for the weak formulation and proves a priori error estimates for the semi-discrete problem by leveraging projection operators with standard approximation and commutation properties. The main result provides an explicit bound at final time in terms of initial-data projection errors and mesh size, demonstrating可靠 convergence of the method. The framework and results are extensible to other conforming FE methods possessing suitable projection operators, enabling rigorous error control for nonlinear electromagnetic simulations.

Abstract

In this paper the semi-discrete finite element approximation of initial boundary value problems for Maxwell's equations in nonliear media of Kerr-type is investigated. For the case of Nédélec elements from the first family, a priori error estimates are established for the approximation.

Semi-discrete finite element approximation applied to Maxwell's equations in nonlinear media

TL;DR

This work analyzes a semi-discrete in space finite element approximation of Maxwell's equations in Kerr-type nonlinear media, using Nédélec elements of the first family. It establishes energy stability for the weak formulation and proves a priori error estimates for the semi-discrete problem by leveraging projection operators with standard approximation and commutation properties. The main result provides an explicit bound at final time in terms of initial-data projection errors and mesh size, demonstrating可靠 convergence of the method. The framework and results are extensible to other conforming FE methods possessing suitable projection operators, enabling rigorous error control for nonlinear electromagnetic simulations.

Abstract

In this paper the semi-discrete finite element approximation of initial boundary value problems for Maxwell's equations in nonliear media of Kerr-type is investigated. For the case of Nédélec elements from the first family, a priori error estimates are established for the approximation.

Paper Structure

This paper contains 8 sections, 4 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Weak formulations
  4. Spatial discretization
  5. Semi-discretization of the weak formulations
  6. The choice of the finite element spaces
  7. An error estimate for the semi-discrete problem
  8. Conclusion

Key Result

Lemma 1.1

Let $\chi^{(1)}, \chi^{(3)}\ge 0$ a.e. in $\Omega$. Then the symmetric matrix $\varepsilon({\mathbf \Psi })$ is uniformly positive definite a.e. for any ${\mathbf \Psi }\in\mathbb{R}^3$.

Theorems & Definitions (12)

  • Lemma 1.1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3
  • Theorem 5.1: Semi-discrete error estimate for the Lee-Madsen formulation
  • ...and 2 more