Convergent adaptive finite element methods for photonic crystal applications. (English) Zbl 1256.65094
Photonic cristals (PC) exhibit interesting properties in the propagation of electromagnetic waves, such as spectral band gaps. In 2D PCs theory, Maxwell’s equations reduce to a two-dimensional one-component wave equation which determines either the electric field (TH mode) or the magnetic field (TE mode). Because of periodicity, one splits each mode into a family of eigenvalue problems which involve non-coercive elliptic operators with discontinuous coefficients. The author extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to PC problems dealing with complications arising essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. The assumptions regarding the conformity and shape regularity of the initial mesh suffice to prove the convergence of the adaptive method in an oscillatory-free way.
The author also presents and proves the convergence of an adaptive method to compute efficiently an entire band in the spectrum. Using this method, the maximum and the minimum of the computed approximation of the band converge to the true global maximum and minimum of the band. The numerical results cover both the cases of periodic structures with and without compact defects. According to the author’s opinion, the paper is the first contribution to the topic of oscillation-free convergence of adaptive finite element methods for PC applications.
The author also presents and proves the convergence of an adaptive method to compute efficiently an entire band in the spectrum. Using this method, the maximum and the minimum of the computed approximation of the band converge to the true global maximum and minimum of the band. The numerical results cover both the cases of periodic structures with and without compact defects. According to the author’s opinion, the paper is the first contribution to the topic of oscillation-free convergence of adaptive finite element methods for PC applications.
Reviewer: Adrian Carabineanu (Bucureşti)
MSC:
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35Q61 | Maxwell equations |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 82D25 | Statistical mechanics of crystals |
| 78A25 | Electromagnetic theory (general) |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
second-order problems; finite element methods; adaptive algorithm; convergence; photonic crystals; oscillation-free; electromagnetic waves; spectral band gaps; Maxwell’s equations; electric field; magnetic field; non-coercive elliptic operators; discontinuous coefficients; elliptic eigenvalue problems; numerical resultsReferences:
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