Courant Institute of Mathematical Sciences

New York University

251 Mercer Street

New York

NY 10012

United States of America

URL: http://www.cims.nyu.edu/~hientzsc

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 *H*^{1}-conforming elements
introduces spurious eigenvalues, and, in general,
the solution of the model problem is not in *H*^{1}.

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.