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

## Summary

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 two- and 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.

**u**ε

_{H}

_{0}(curl,

*Ω*) such that for all

**v**ε

*H*

_{0}(curl,

*Ω*)

Here, *Ω* is a bounded, open, connected polyhedron in \(\mathbb{R}\)^{3} or a polygon in \(\mathbb{R}\)^{2}, *H*(curl*Ω*) is the space of vectors in (*L*^{2}(*Ω*))^{2} or (*L*^{2}(*Ω*))^{3} with curl in *L*^{2}(*Ω*) or (*L*^{2}(*Ω*))^{3}, respectively; *H*_{0}(curl *Ω*) is its subspace of vectors with vanishing tangential components on *∂Ω*; **f** ε (*L*^{2}(*Ω*))^{d} for *d* = 2, 3, and (·, ·) denotes the inner product in *L*^{2}(*Ω*) of functions or vector fields. For simplicity, we will assume that *α* and *β* are piecewise constant.

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