# An Iterative Substructuring Algorithm for Two-Dimensional Problems in H(curl)

@article{Dohrmann2012AnIS, title={An Iterative Substructuring Algorithm for Two-Dimensional Problems in H(curl)}, author={Clark R. Dohrmann and Olof B. Widlund}, journal={SIAM J. Numer. Anal.}, year={2012}, volume={50}, pages={1004-1028} }

A domain decomposition algorithm, similar to classical iterative substructuring algorithms, is presented for two-dimensional problems in the space $H_0(\mbox{curl};\Omega)$. It is defined in terms of a coarse space and local subspaces associated with individual edges of the subdomains into which the domain of the problem has been subdivided. The algorithm differs from others in three basic respects. First, it can be implemented in an algebraic manner that does not require access to individual…

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## 32 Citations

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## References

SHOWING 1-10 OF 41 REFERENCES

An Analysis of a FETI-DP Algorithm on Irregular Subdomains in the Plane

- Computer ScienceSIAM J. Numer. Anal.
- 2008

Borders are derived for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain.

Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions

- Computer ScienceSIAM J. Numer. Anal.
- 2008

In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented and it is shown that the condition number of the domain decomposition method is bounded by $C(1+H/\delta)(1+\log(H/h))^2$, where the constant C is independent of the number of subDomains and possible jumps in coefficients betweenSubdomains.

An iterative substructuring method for Maxwell's equations in two dimensions

- Computer ScienceMath. Comput.
- 2001

The same result is established for similar iterative methods for low--order N{\''e}d{\'' e}lec finite elements, which approximate $\Hcurl$ in two dimensions.

Domain Decomposition Methods for Raviart-Thomas Vector Fields

- Computer Science
- 2011

Two domain decomposition methods for solving vector field problems posed in H(div) discretized by Raviart-Thomas finite elements are introduced and a two-level overlapping Schwarz method is developed.

An Overlapping Schwarz Algorithm for Almost Incompressible Elasticity

- Computer ScienceSIAM J. Numer. Anal.
- 2009

Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces and a bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains.

A family of energy minimizing coarse spaces for overlapping schwarz preconditioners

- Computer Science
- 2008

A simple and effective approach is presented to construct coarse spaces for overlapping Schwarz preconditioners. The approach is based on energy minimizing extensions of coarse trace spaces, and can…

Dual-primal FETI algorithms for edge finite-element approximations in 3D

- Computer Science
- 2006

A family of dual-primal finite-element tearing and interconnecting methods for edge-element approximations in 3D is proposed and analysed and a strong coupling between degrees of freedom associated with subdomain edges and faces is relied on.

A FETI Domain Decomposition Method for Edge Element Approximations in Two Dimensions with Discontinuous Coefficients

- Computer ScienceSIAM J. Numer. Anal.
- 2001

A class of finite element tearing and interconnecting (FETI) methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed and it is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with theNumber of unknowns associated with individual substructure.

Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces

- Computer ScienceSIAM J. Numer. Anal.
- 2007

This paper develops and analyzes a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div,)-elliptic variational problems and proves mesh-independent effectivity of the precondITIONers by using the abstract theory of auxiliary space preconditionsing.

Neumann-Neumann Methods for Vector Field Problems

- Mathematics
- 1999

In this paper, we study some Schwarz methods of Neumann-Neumann type for some vector field problems, discretized with the lowest order Raviart-Thomas and Nedelec finite elements. We consider a hybrid…