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On stabilized P1 finite element approximation fortime harmonic Maxwell's equations

M. Asadzadeh, L. Beilina

TL;DR

This paper is devoted to the study of the stabilized linear finite element method for the time harmonic Maxwell's equations in a dual form obtained through the Laplace transformation in time.

Abstract

One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations, is to render them from their hyperbolic character to elliptic form. This paper is devoted to the study of the stabilized linear finite element method for the time harmonic Maxwell's equations in a dual form obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. For the stabilized model a coercivity relation is derived that guarantee's the existence of a unique solution for the iscrete problem. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space and hence optimal. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residual of the computed solution. The posteriori approach is used for constructing adaptive algorithms for the computational purposes. Further, assuming a sufficiently regular solution for the dual problem, we reach the same convergence of O(h). Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme.

On stabilized P1 finite element approximation fortime harmonic Maxwell's equations

TL;DR

This paper is devoted to the study of the stabilized linear finite element method for the time harmonic Maxwell's equations in a dual form obtained through the Laplace transformation in time.

Abstract

One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations, is to render them from their hyperbolic character to elliptic form. This paper is devoted to the study of the stabilized linear finite element method for the time harmonic Maxwell's equations in a dual form obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. For the stabilized model a coercivity relation is derived that guarantee's the existence of a unique solution for the iscrete problem. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space and hence optimal. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residual of the computed solution. The posteriori approach is used for constructing adaptive algorithms for the computational purposes. Further, assuming a sufficiently regular solution for the dual problem, we reach the same convergence of O(h). Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme.

Paper Structure

This paper contains 12 sections, 4 theorems, 72 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The mathematical model
  3. Variational approach
  4. Stabilized model
  5. Finite element discretization
  6. Error analysis
  7. A priori error estimates
  8. A posteriori error estimates
  9. A duality approach
  10. Adaptivity algorithm
  11. Numerical examples
  12. Conclusion

Key Result

Theorem 1

The problem eq6 has a unique solution $\hat{E}_h\in {\textbf{W}}_h^E(\Omega)$.

Figures (8)

  • Figure 1: Domain decomposition in $\Omega$.
  • Figure 2: The function $\varepsilon(x,y)$ in the domain $\Omega=[0,1] \times[0,1]$ for different $m$ in \ref{['eps']}.
  • Figure 3: Relative errors for $m=2$ (left) and $m=3$ (right).
  • Figure 4: Relative errors for $m=4$ (left) and $m=5$ (right).
  • Figure 5: Relative errors for $m=6$ (left) and $m=7$ (right).
  • ...and 3 more figures

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Theorem 1: well-posedness
  • proof
  • Theorem 2
  • proof
  • Remark 4.1
  • Theorem 3
  • proof
  • ...and 2 more