One fundamental difficulty in low frequency electromagnetics computation is the low-frequency breakdown problem. It makes the discretized linear system very poorly conditioned and thus difficult to solve. This issue is present in both integral equation and partial differential equation solution methods, and thus has attracted many researchers who have proposed various methods to overcome this difficulty. In this work, we propose a novel mixed spectral element method (mixed SEM) to alleviate the low frequency breakdown problem and apply this new method to solve low-frequency subsurface sensing problems.
The traditional spectral element method (SEM) has been shown in recent years to produce spectral and high-order accuracy in wave propagation and scattering problems. By using the spectral edge elements, the SEM has successfully removed the non-zero spurious modes, as in the traditional finite element method (FEM) for electromagnetics. It is also well known that, however, zero (DC) spurious modes still exist in both FEM and SEM, and such spurious modes may corrupt or degrade the numerical solutions, especially for the eigenvalue problems in waveguides and cavities. These spurious modes are confined to the subspace of zero eigenvalues, making it difficult to determine which modes are physical. Furthermore, these spurious modes also slow down the convergence of desirable modes and cause additional computational complications. In our recent work, we developed a novel mixed SEM to remove such zero spurious modes and significantly improve the numerical solutions by enforcing the divergence free condition (i.e., Gauss' law) in the SEM (N. Liu et al., Commun. Comput. Phys., 17, 458-486, 2015; N. Liu et al., IEEE Trans. Microwave Theory Tech., 63, 317-325, 2015). This mixed SEM is actually also capable of solving the low frequency breakdown problem, because Gauss’ law is now explicitly enforce, making system matrix well-conditioned even at extremely low frequency. In this presentation, we will demonstrate this mixed SEM with the applications in cavity and waveguide problems, and also low frequency electromagnetic sensing of subsurface objects.
