The goal of this project is to develop efficient algorithms for the numerical
solution of problems arising from various electromagnetic models.
Specifically, we are interested in the system of Maxwell's equations and the
related eigenvalue problem.
These are important in practical applications, where one needs to compute the
electromagnetic field generated by prescribed current and charges, or in the
computation of the eigenmodes that will propagate through a given medium.
Our approach is based on a very weak variational formulations of the div-curl
systems corresponding to the electrostatic and magnetostatic problems.
The finite dimensional approximation is a negative norm finite element
least-squares algorithm which uses different solution and test spaces.
This allows for approximation of problems with low regularity, where the
solution is only in L2 and the data resides in various dual spaces.
The solution operators for the above problems are further used to obtain an
approximation to the eigenvalue problem.
The resulting discretization method has the advantages of avoiding potentials
and the use of Nedelec spaces.
In fact, we allow for the mixing of continuous and discontinuous approximation
spaces of varying polynomial degrees.
Additional advantages are that the matrix of the discrete system is
uniformly equivalent to the mass matrix, and that spurious eigenmodes are
completely avoided.
Finally, the dual inner products can be efficiently implemented using
preconditioner for standard second order problems (for example a sweep of
multigrid).
