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

  • Bernhard Hientzsch
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


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.

The model problem is: Find u ε H0(curl, Ω) such that for all v ε H0 (curl, Ω)
$$a(u,v): = (\alpha u,v) + (\beta CURLu,CURLv) = (f,v)$$

Here, Ω is a bounded, open, connected polyhedron in \(\mathbb{R}\)3 or a polygon in \(\mathbb{R}\)2, H(curlΩ) is the space of vectors in (L2(Ω))2 or (L2(Ω))3 with curl in L2(Ω) or (L2(Ω))3, respectively; H0(curl Ω) is its subspace of vectors with vanishing tangential components on ∂Ω; f ε (L2(Ω))d for d = 2, 3, and (·, ·) denotes the inner product in L2(Ω) of functions or vector fields. For simplicity, we will assume that α and β are piecewise constant.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Bernhard Hientzsch
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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