Compatible Subdomain Level Isotropic/Anisotropic Discontinuous Galerkin Time Domain (DGTD) Method for Multiscale Simulation

Domain decomposition method provides a solution for the very large electromagnetic

system which are impossible for single domain methods. Discontinuous Galerkin

(DG) method can be viewed as an extreme version of the domain decomposition,

i.e., each element is regarded as one subdomain. The whole system is solved element

by element, thus the inversion of the large global system matrix is no longer necessary,

and much larger system can be solved with the DG method compared to the

continuous Galerkin (CG) method.

In this work, the DG method is implemented on a subdomain level, that is, each subdomain contains multiple elements. The numerical flux only applies on the

interfaces between adjacent subdomains. The subodmain level DG method divides

the original large global system into a few smaller ones, which are easier to solve,

and it also provides the possibility of parallelization. Compared to the conventional

element level DG method, the subdomain level DG has the advantage of less total

DoFs and fexibility in interface choice. In addition, the implicit time stepping is

relatively much easier for the subdomain level DG, and the total CPU time can be

much less for the electrically small or multiscale problems.

The hybrid of elements are employed to reduce the total DoF of the system.

Low-order tetrahedrons are used to catch the geometry ne parts and high-order

hexahedrons are used to discretize the homogeneous and/or geometry coarse parts.

In addition, the non-conformal mesh not only allow dierent kinds of elements but

also sharp change of the element size, therefore the DoF can be further decreased.

The DGTD method in this research is based on the EB scheme to replace the

previous EH scheme. Dierent from the requirement of mixed order basis functions

for the led variables E and H in the EH scheme, the EB scheme can suppress the

spurious modes with same order of basis functions for E and B. One order lower in

the basis functions in B brings great benets because the DoFs can be signicantly

reduced, especially for the tetrahedrons parts.

With the basis functions for both E and B, the EB scheme upwind

ux and

EB scheme Maxwellian PML, the eigen-analysis and numerical results shows the

eectiveness of the proposed DGTD method, and multiscale problems are solved

eciently combined with the implicit-explicit hybrid time stepping scheme and multiple

kinds of elements.

The EB scheme DGTD method is further developed to allow arbitrary anisotropic

media via new anisotropic EB scheme upwind

ux and anisotropic EB scheme

Maxwellian PML. The anisotropic M-PML is long time stable and absorb the outgoing

wave eectively. A new TF/SF boundary condition is brought forward to

simulate the half space case. The negative refraction in YVO4 bicrystal is simulated

with the anisotropic DGTD and half space TF/SF condition for the rst time with

numerical methods.