A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, p. 213-251
We propose a new reduced basis element-cum-component mode synthesis approach for parametrized elliptic coercive partial differential equations. In the Offline stage we construct a Library of interoperable parametrized reference components relevant to some family of problems; in the Online stage we instantiate and connect reference components (at ports) to rapidly form and query parametric systems. The method is based on static condensation at the interdomain level, a conforming eigenfunction “port” representation at the interface level, and finally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at the intradomain level. We show under suitable hypotheses that the RB Schur complement is close to the FE Schur complement: we can thus demonstrate the stability of the discrete equations; furthermore, we can develop inexpensive and rigorous (system-level) a posteriori error bounds. We present numerical results for model many-parameter heat transfer and elasticity problems with particular emphasis on the Online stage; we discuss flexibility, accuracy, computational performance, and also the effectivity of the a posteriori error bounds.
DOI : https://doi.org/10.1051/m2an/2012022
Classification:  35J25,  65N30,  65D99
Keywords: reduced basis method, reduced basis element method, domain decomposition, Schur complement, elliptic partial differential equations, a posteriori error estimation, component mode synthesis, parametrized systems
     author = {Phuong Huynh, Dinh Bao and Knezevic, David J. and Patera, Anthony T.},
     title = {A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {213-251},
     doi = {10.1051/m2an/2012022},
     zbl = {1276.65082},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_1_213_0}
Phuong Huynh, Dinh Bao; Knezevic, David J.; Patera, Anthony T. A Static condensation Reduced Basis Element method : approximation and a posteriori error estimation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 213-251. doi : 10.1051/m2an/2012022. http://www.numdam.org/item/M2AN_2013__47_1_213_0/

[1] H. Antil, M. Heinkenschloss, R.H.W. Hoppe and D.C. Sorensen, Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. Comput. Visualization Sci. 13 (2010) 249-264. | MR 2748450 | Zbl 1220.65074

[2] H. Antil, M. Heinkenschloss and R.H.W. Hoppe, Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. Optim. Methods Softw. 26 (2011) 643-669, doi: 10.1080/10556781003767904. | MR 2837792 | Zbl 1227.49046

[3] 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. | MR 2086832 | Zbl 1133.65304

[4] A. Bermúdez and F. Pena, Galerkin lumped parameter methods for transient problems. Int. J. Numer. Methods Eng. 87 (2011) 943-961, doi: 10.1002/nme.3140. | MR 2835769 | Zbl 1242.80005

[5] P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. Technical Report, Aachen Institute for Advanced Study in Computational Engineering Science, preprint : AICES-2010/05-2 (2010). | MR 2821591 | Zbl 1229.65193

[6] F. Bourquin, Component mode synthesis and eigenvalues of second order operators : discretization and algorithm. ESAIM : M2AN 26 (1992) 385-423. | Numdam | MR 1160133 | Zbl 0765.65100

[7] S.C. Brenner, The condition number of the Schur complement in domain decompostion. Numer. Math. 83 (1999) 187-203. | MR 1712684 | Zbl 0936.65141

[8] A. Buffa, Y. Maday, A.T. Patera, C. Prud'Homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis. To appear in ESAIM : M2AN (2010). | Numdam | Zbl 1272.65084

[9] Y. Chen, J.S. Hesthaven and Y. Maday, A Seamless Reduced Basis Element Methods for 2D Maxwell's Problem : An Introduction, edited by J. Hesthaven and E.M. Rønquist, in Spectral and High Order Methods for Partial Differential Equations-Selected papers from the ICASOHOM'09 Conference 76 (2011). | MR 3204811 | Zbl 1217.78056

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

[11] J.L. Eftang, D.B.P. Huynh, D.J. Knezevic, E.M. Rønquist and A.T. Patera, Adaptive port reduction in static condensation, in MATHMOD 2012 - 7th Vienna International Conference on Mathematical Modelling (2012) (Submitted).

[12] M. Ganesh, J.S. Hesthaven and B. Stamm, A reduced basis method for multiple electromagnetic scattering in three dimensions. Technical Report 2011-9, Scientific Computing Group, Brown University, Providence, RI, USA (2011).

[13] G. Golub and C. Van Loan, Matrix Computations. Johns Hopkins University Press (1996). | MR 1417720 | Zbl 0865.65009

[14] B. Haggblad and L. Eriksson, Model reduction methods for dynamic analyses of large structures. Comput. Struct. 47 (1993) 735-749. | MR 1224096 | Zbl 0811.73062

[15] U.L. Hetmaniuk and R.B. Lehoucq, A special finite element method based on component mode synthesis. ESAIM : M2AN 44 (2010) 401-420. | Numdam | MR 2666649 | Zbl 1190.65173

[16] T.Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | MR 1455261 | Zbl 0880.73065

[17] W.C. Hurty, On the dynamic analysis of structural systems using component modes, in First AIAA Annual Meeting. Washington, DC, AIAA paper, No. 64-487 (1964).

[18] D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345 (2007) 473-478. | MR 2367928 | Zbl 1127.65086

[19] L. Iapichino, Quarteroni and G.A., Rozza, A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks. Comput. Methods Appl. Mech. Eng. 221-222 (2012) 63-82. | MR 2913950 | Zbl 1253.76139

[20] H. Jakobsson, F. Beingzon and M.G. Larson, Adaptive component mode synthesis in linear elasticity. Int. J. Numer. Methods Eng. 86 (2011) 829-844. | MR 2829770 | Zbl 1235.74288

[21] S. Kaulmann, M. Ohlberger and B. Haasdonk, A new local reduced basis discontinuous galerkin approach for heterogeneous multiscale problems. C. R. Math. 349 (2011) 1233-1238. | MR 2861991 | Zbl 1269.65127

[22] B.S. Kirk, J.W. Peterson, R.H. Stogner and G.F. Carey, libMesh : A C++ library for Parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22 (2006) 237-254.

[23] D.J. Knezevic and J.W. Peterson, A high-performance parallel implementation of the certified reduced basis method. Comput. Methods Appl. Mech. Eng. 200 (2011) 1455-1466. | Zbl 1228.76109

[24] Y Maday and EM Rønquist, The reduced basis element method : Application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004) 240-258. | MR 2114342 | Zbl 1077.65120

[25] Y. Maday, A.T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437-446. | MR 1910581 | Zbl 1014.65115

[26] N.C. Nguyen, A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys. 227 (2007) 9807-9822. | MR 2469035 | Zbl 1155.65391

[27] C. Prud'Homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations : Reduced-basis output bounds methods. J. Fluids Eng. 124 (2002) 70-80.

[28] G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR 2430350 | Zbl pre05344486