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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)

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.

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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References

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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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