STEFANO GIANI
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A Posteriori Error Estimator for Application with Photonic Crystals

 

We derived an a posteriori error estimator for hp-adaptive continuous Galerkin methods for photonic crystal (PC) applications.
Our goal was to compute efficiently the behaviour of light inside PC devices.
The error estimator that we propose is based on the residual of the discrete problem and we show that it leads to very fast convergence in all considered examples when used with hp-adaptive refinement techniques. We proved that our error estimator is reliable and efficient.

Line Defect:

Line Defect Figure 1. Photonic crystal with a line defect

In Fig. 1 we have an example of a photonic crystal with a line defect. This type of configurations can be used to study PC waveguide.
In Fig. 2 we present a TE eigenfunction trapped in the the gap of the structure and in Fig. 3 the corresponding hp-adapted mesh.
As can be seen the method heavily refine in h around the corners of the inclusions where the eigenfunction has singularities in the gradient.

Solution
Figure 2. TE Eigenfunction trapped in the waveguide
     
Adapted Mesh Figure 3. hp-adapted mesh for the trapped mode

 

V-bend Crystal:

V-bend Crystal Figure 4. Photonic crystal with a V-bend

The trapped modes in the bend are important in practice because exciting these modes it is possible to make an electromagnetic wave to go around a bend.
In Fig. 4 we have an example of a photonic crystal with a V-bend. This type of configurations can be used to study PC waveguide with bends.
In Fig. 5 we present a TE eigenfunction trapped in the the bend of the structure and in Fig. 6 the corresponding hp-adapted mesh.
As can be seen the method heavily refine in h around the corners of the inclusions where the eigenfunction has singularities in the gradient.

Solution
Figure 5. TE Eigenfunction trapped in the bend
     
Adapted Mesh Figure 6. hp-adapted mesh for the trapped mode

 

Surface of a PC:

Surface Figure 7. Surface of a photonic crystal

In this section we approximate a mode localized on the surface of a semi-infinite PC.
In Fig. 7 we have an example of a the surface of a photonic crystal. This type of configurations can be used to study PC waveguide.
In Fig. 8 we present a TE eigenfunction trapped on the surface of the crystal and in Fig. 9 the corresponding hp-adapted mesh.
As can be seen the method heavily refine in h around the corners of the inclusions where the eigenfunction has singularities in the gradient.

Solution
Figure 8. TE Eigenfunction trapped on the surface
     
Adapted Mesh Figure 9. hp-adapted mesh for the trapped mode

 

References:

S. Giani (2013), An a posteriori error estimator for hp-adaptive continuous Galerkin methods for photonic crystal applications. - Computing 95(5), 395-414.

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