A special finite element method based on component mode synthesis
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, p. 401-420

The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients. The proposed basis functions are inspired by the classic idea of component mode synthesis and exploit an orthogonal decomposition of the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.

DOI : https://doi.org/10.1051/m2an/2010007
Classification:  35J20,  65F15,  65N25,  65N30,  65N55
Keywords: eigenvalues, modal analysis, multilevel, substructuring, domain decomposition, dimensional reduction, finite elements
@article{M2AN_2010__44_3_401_0,
     author = {Hetmaniuk, Ulrich L. and Lehoucq, Richard B.},
     title = {A special finite element method based on component mode synthesis},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {3},
     year = {2010},
     pages = {401-420},
     doi = {10.1051/m2an/2010007},
     zbl = {1190.65173},
     mrnumber = {2666649},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_3_401_0}
}
Hetmaniuk, Ulrich L.; Lehoucq, Richard B. A special finite element method based on component mode synthesis. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, pp. 401-420. doi : 10.1051/m2an/2010007. http://www.numdam.org/item/M2AN_2010__44_3_401_0/

[1] I. Babuška and J.E. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510-536. | Zbl 0528.65046

[2] I. Babuška, G. Caloz and J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31 (1994) 945-981. | Zbl 0807.65114

[3] I. Babuška, U. Banerjee and J. Osborn, On principles for the selection of shape functions for the generalized finite element method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 5595-5629. | Zbl 1016.65052

[4] I. Babuška, U. Banerjee and J.E. Osborn, Generalized finite element methods - main ideas, results and perspective. Int. J. Comp. Meths. 1 (2004) 67-103. | Zbl 1081.65107

[5] J.K. Bennighof and R.B. Lehoucq, An automated multilevel substructuring method for eigenspace computation in linear elastodynamics. SIAM J. Sci. Comput. 25 (2004) 2084-2106. | Zbl 1133.65304

[6] F. Bourquin, Component mode synthesis and eigenvalues of second order operators: Discretization and algorithm. ESAIM: M2AN 26 (1992) 385-423. | Numdam | Zbl 0765.65100

[7] F. Brezzi and L. Marini, Augmented spaces, two-level methods, and stabilizing subgrids. Int. J. Numer. Meth. Fluids 40 (2002) 31-46. | Zbl 1021.76024

[8] R.R. Craig, Jr. and M.C.C. Bampton, Coupling of substructures for dynamic analysis. AIAA J. 6 (1968) 1313-1319. | Zbl 0159.56202

[9] Y. Efendiev and T. Hou, Multiscale Finite Element Methods: Theory and Applications, Surveys and Tutorials in the Applied Mathematical Sciences 4. Springer, New York, USA (2009). | Zbl 1163.65080

[10] U. Hetmaniuk and R.B. Lehoucq, Multilevel methods for eigenspace computations in structural dynamics, in Domain Decomposition Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 55, Springer-Verlag (2007) 103-114.

[11] T. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | Zbl 0880.73065

[12] W.C. Hurty, Vibrations of structural systems by component-mode synthesis. J. Eng. Mech. Division ASCE 86 (1960) 51-69.

[13] J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems. Multiscale Model. Simul. 7 (2008) 171-196. | Zbl 1160.65342

[14] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations - Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, UK (1999). | Zbl 0931.65118