In recent years, high-order discretizations for Maxwell's equations and related problems have been designed. In our contribution, we will consider spectral element type discretizations of the following model problem in H(curl):
which appears in the implicit time-stepping of the second-order Maxwell evolution equations, and in the time-harmonic approach.
The discretization of the model problem requires H(curl)-conforming elements, since the use of H1-conforming elements introduces spurious eigenvalues, and, in general, the solution of the model problem is not in H1.
For problems in computational fluid dynamics, spectral element discretizations have been very successful. They combine superior approximation properties, geometric flexibility and a special structure that can be exploited to construct fast algorithms. We extend the spectral element method to our model problem, still using nodal degrees of freedom associated with tensor Gauss-Lobatto-Legendre grids, while enforcing only the continuity of the tangential components across element interfaces. (In this way, we span the same global space as a generalized hN-extension of the Nédélec edge elements.)
We develop fast direct solvers for these spectral Nédélec element discretizations in rectangular domains. Using these solvers as local solvers, we implement overlapping Schwarz methods for the model problem in two dimensions, and present extensive numerical tests and an analytic condition number estimate. This estimate extends the theory in Toselli (Numerische Mathematik, 86(4):733-752, 2000) to the spectral element case. We reduce the proof of the condition number estimate to three basic estimates, which we discuss both analytically and numerically. Our proof is valid in both two and three dimensions.