# 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.

## References

1. F. B. Belgacem and C. Bernardi. Spectral element discretization of the Maxwell equations. Math. Comp., 68(228):1497–1520, 1999.
2. B. Hientzsch. Fast Solvers and Domain Decomposition Preconditioners for Spectral Element Discretizations of Problems in H(curl). PhD thesis, Courant Institute of Mathematical Sciences, September 2001. Technical Report TR2001-823, Department of Computer Science, Courant Institute.Google Scholar
3. B. Hientzsch. Overlapping Schwarz preconditioners for spectral Nédélec elements for a model problem in H(curl). Technical Report TR2002-834, Department of Computer Science, Courant Institute of Matical Sciences, November 2002.Google Scholar
4. B. Hientzsch. Fast solvers and preconditioners for spectral Nédélec element discretizations of a model problem in H(curl). In I. Herrera, D. E. Keyes, O. B. Widlund, and R. Yates, editors, Domain Decomposition Methods in Science and Engineering, pages 427–433. National Autonomous University of Mexico (UNAM), Mexico City, Mexico, 2003. Proceedings of the 14th International Conference on Domain Decomposition Methods in Cocoyoc, Mexico, January 6–11, 2002.Google Scholar
5. P. Monk. On the p-and hp-extension of Nédélec's curl-conforming elements. J. Comput. Appl. Math., 53(1):117–137, 1994. ISSN 0377-0427.
6. J.-C. Nédélec. Mixed finite elements in R3. Numer. Math., 35:315–341, 1980.
7. J.-C. Nédélec. A new family of mixed finite elements in R3. Numer. Math., 50:57–81, 1986.
8. B. F. Smith, P. E. Bjørstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996.Google Scholar