Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions

In this paper, we present several domain decomposition preconditioners for high-order Spectral Nédélec element discretizations for a Maxwell model problem in H(curl), in particular overlapping Schwarz preconditioners and Balancing Neumann-Neumann preconditioners. For an efficient and fast implementation of these preconditioners, fast matrix-vector products and direct solvers for problems posed on one element or a small array of elements are needed. In previous work, we have presented such algorithms for the two-dimensional case; here, we will present a new fast solver that works both in the twoand three-dimensional case. Next, we define the preconditioners considered in this paper, present numerical results for overlapping methods in three dimensions and Balancing Neumann-Neumann methods in two dimensions. We will also give a condition number estimate for the overlapping Schwarz method. The model problem is: Find u ∈ H0(curl, Ω) such that for all v ∈ H0(curl, Ω)