FETI

In mathematics, in particular numerical analysis, the FETI method (finite element tearing and interconnect) is an iterative substructuring method for solving systems of linear equations from the finite element method for the solution of elliptic partial differential equations, in particular in computational mechanics[1] In each iteration, FETI requires the solution of a Neumann problem in each substructure and the solution of a coarse problem. The simplest version of FETI with no preconditioner (or only a diagonal preconditioner) in the substructure is scalable with the number of substructures[2] but the condition number grows polynomially with the number of elements per substructure. FETI with a (more expensive) preconditioner consisting of the solution of a Dirichlet problem in each substructure is scalable with the number of substructures and its condition number grows only polylogarithmically with the number of elements per substructure.[3] The coarse space in FETI consists of the nullspace on each substructure. Apart from FETI Dual-Primal (FETI-DP, see below), several extensions have been developed to solve particular physical problems, as FETI Helmholtz (FETI-H),[4][5] FETI for quasi-incompressible problems,[6] and FETI Contact (FETI-C).[7][8][9]

## See also

* Balancing domain decomposition
* FETI-DP

## References

1. ↑ C. Farhat and F. X. Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Internat. J. Numer. Meths. Engrg. 32, 1205-1227 (1991)
2. ↑ Charbel Farhat, Jan Mandel, and François-Xavier Roux, Optimal convergence properties of the FETI domain decomposition method, Comput. Meth. Appl. Mech. Engrg. 115(1994)365-385
3. ↑ J. Mandel and R. Tezaur, On the Convergence of a Substructuring Method with Lagrange multipliers, Numerische Mathematik 73 (1996) 473-487
4. ↑ C. Farhat, A. Macedo, M. Lesoinne, A two-level domain decom- position method for the iterative solution of high-frequency exterior Helmholtz problems, Numerische Mathematik 85 (2000) 283-303 DOI 10.1007/PL00005389
5. ↑ C. Farhat, A. Macedo, M. Lesoinne, F. X. Roux, F. Magoules, A. D. L. Bourdonnaye, Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems, Computer Methods in Applied Mechanics and Engineering 184(2-4) (2000) 213-240 DOI 10.1016/s0045-7825(99)00229-7 hal-00624498
6. ↑ B. Vereecke, H. Bavestrello, D. Dureisseix, An extension of the FETI domain decomposition method for incompressible and nearly incompressible problems, Computer Methods in Applied Mechanics and Engineering 192 (2003) 3409-3429 DOI 10.1016/S0045-7825(03)00313-X hal-00141163
7. ↑ D. Dureisseix, C. Farhat, A numerically scalable domain decomposition method for the solution of frictionless contact problems, International Journal for Numerical Methods in Engineering 50 (2001) 2643-2666 DOI 10.1002/nme.140 hal-00321391
8. ↑ Z. Dostál, F. A.M. Gomes Neto, S. A. Santos, Solution of contact problems by FETI domain decomposition with natural coarse space projections. Computer Methods in Applied Mechanics and Engineering 190 (2000) 1611-1627 DOI 10.1016/s0045-7825(00)00180-8
9. ↑ Philip Avery, Charbel Farhat, The FETI family of domain decomposition methods for inequality-constrained quadratic programming: Application to contact problems with conforming and nonconforming interfaces. Computer Methods in Applied Mechanics and Engineering 198 (2009) 1673-1683, DOI 10.1016/j.cma.2008.12.014

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