A Null Space Free Jacobi-Davidson Iteration for Maxwell's Operator

We present an efficient null space free Jacobi-Davidson method to compute the positive eigenvalues of the degenerate elliptic operator arising from Maxwell’s equations. We consider spatial compatible discretizations such as Yee’s scheme which guarantee the existence of a discrete vector potential. During the Jacobi-Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in original Jacobi-Davidson approach. The computational cost of the transformation from the vector potential to the corrector is negligible. As a consequence, the expanding subspace automatically stays out of the null space and no extra projection step is needed. Numerical evidence confirms that the new method is much more efficient than the original Jacobi-Davidson method.