×

Adaptive reduced basis finite element heterogeneous multiscale method. (English) Zbl 1286.74088

Summary: An adaptive reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is proposed for elliptic problems with multiple scales. The multiscale method is based on the RB-FE-HMM introduced in [A. Abdulle and Y. Bai, J. Comput. Phys. 231, No. 21, 7014–7036 (2012; Zbl 1284.65161)]. It couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains. Unlike classical numerical homogenization methods, the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions (the reduced basis) obtained by a greedy algorithm in an offline stage. In this paper we present a residual-based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest. For both type of adaptive strategies, rigorous a posteriori error estimates are derived and corresponding error estimators are proposed. In contrast to the adaptive finite element heterogeneous multiscale method (FE-HMM), there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement. Up to an offline preliminary stage, the RB-FE-HMM has the same computational complexity as a standard adaptive FEM for the effective problem. Two and three dimensional numerical experiments confirm the efficiency of the RB-FE-HMM and illustrate the improvements compared to the adaptive FE-HMM.

MSC:

74Q20 Bounds on effective properties in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure

Citations:

Zbl 1284.65161
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Oden, J.; Prudhomme, S.; Romkes, A.; Bauman, P., Multiscale modeling of physical phenomena: adaptive control of models, SIAM J. Sci. Comput., 28, 6, 2359-2389 (2006) · Zbl 1126.74006
[2] Abdulle, A., Analysis of a heterogeneous multiscale FEM for problems in elasticity, Math. Models Methods Appl. Sci., 16, 4, 615-635 (2006) · Zbl 1095.74029
[3] Efendiev, Y.; Hou, T. Y., Multiscale finite element methods. Theory and applications, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4 (2009), Springer: Springer New York · Zbl 1163.65080
[4] Geers, M.; Kouznetsova, V.; Brekelmans, W., Multi-scale computational homogenization: trends and challenges, J. Comput. Appl. Math., 234, 2175-2182 (2010) · Zbl 1402.74107
[5] Abdulle, A.; E, W.; Engquist, B.; Vanden-Eijnden, E., The heterogeneous multiscale method, Acta Numer., 21, 1-87 (2012) · Zbl 1255.65224
[6] E, W.; Engquist, B., The heterogeneous multiscale methods, Commun. Math. Sci., 1, 1, 87-132 (2003) · Zbl 1093.35012
[7] Abdulle, A., The finite element heterogeneous multiscale method: a computational strategy for multiscale pdes, GAKUTO Int. Ser. Math. Sci. Appl., 31, 135-184 (2009) · Zbl 1372.65266
[8] Abdulle, A., On a priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM Multiscale Model. Simul., 4, 2, 447-459 (2005) · Zbl 1092.65093
[9] Abdulle, A.; Bai, Y., Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys., 231, 21, 7014-7036 (2012) · Zbl 1284.65161
[10] Prud’homme, C.; Rovas, D. V.; Veroy, K.; Machiels, L.; Maday, Y.; Patera, A. T.; Turinici, G., Reliable real-time solution of parametrized partial differential equations: reduced-basis output bounds methods, J. Fluids Engrg., 124, 70-80 (2002)
[12] Rozza, G.; Huynh, D.; Patera, A. T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods. Engrg., 15, 229-275 (2008) · Zbl 1304.65251
[13] Boyaval, S., Reduced-basis approach for homogenization beyond the periodic setting, Multiscale Model. Simul., 7, 1, 466-494 (2008) · Zbl 1156.74358
[15] Ainsworth, M.; Oden, J., A Posteriori Error Estimation in Finite Element Analysis (2000), John Wiley & Sons: John Wiley & Sons New york · Zbl 1008.65076
[16] Verfürth, R., A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996), Wiley-Teubner: Wiley-Teubner New-York · Zbl 0853.65108
[17] Oden, J.; Prudhomme, S., Goal-oriented error estimation and adaptivity for the finite element method, Comput. Math. Appl., 41, 5-6, 735-756 (2001) · Zbl 0987.65110
[18] Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10, 1-102 (2001) · Zbl 1105.65349
[19] Ainsworth, M.; Rankin, R., Guaranteed computable bounds on quantities of interest in finite element computations, Int. J. Numer. Methods Engrg., 89, 13, 1605-1634 (2012) · Zbl 1242.65232
[20] Moorthy, S.; Ghosh, S., Adaptivity and convergence in the Voronoi cell nite element model for analyzing heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 185, 185, 37-74 (2000) · Zbl 0967.74068
[21] Moorthy, S.; Ghosh, S., Concurrent multi-scale analysis of elastic composites by a multi-level computational model, Comput. Methods Appl. Mech. Engrg., 193, 497-538 (2004) · Zbl 1060.74590
[22] Zohdi, T. I.; Oden, J. T.; Rodin, G. J., Hierarchical modeling of heterogeneous bodies, Comput. Methods Appl. Mech. Engrg., 138, 1-4, 273-298 (1996) · Zbl 0921.73080
[23] Oden, J. T.; Vemaganti, K. S., Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials: I. Error estimates and adaptive algorithms, J. Comput. Phys., 164, 1, 22-47 (2000) · Zbl 0992.74072
[24] Ohlberger, M., A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Model. Simul, 4, 1, 88-114 (2005) · Zbl 1090.65128
[25] Nguetseng, G., A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20, 3, 608-623 (1989) · Zbl 0688.35007
[26] Abdulle, A.; Nonnenmacher, A., Adaptive finite element heterogeneous multiscale method for homogenization problems, Comput. Methods Appl. Mech. Engrg., 200, 37-40, 2710-2726 (2011) · Zbl 1230.74165
[28] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic Analysis for Periodic Structures (1978), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0411.60078
[29] Jikov, V.; Kozlov, S.; Oleinik, O., Homogenization of Differential Operators and Integral Functionals (1994), Springer-Verlag: Springer-Verlag Berlin, Heidelberg
[30] Ciarlet, P.; Raviart, P., The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, Math. Found. FEM Appl. PDE, 409-474 (1972) · Zbl 0262.65070
[31] E, W.; Ming, P.; Zhang, P., Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18, 1, 121-156 (2005) · Zbl 1060.65118
[32] Abdulle, A., A priori and a posteriori error analysis for numerical homogenization: a unified framework, Ser. Contemp. Appl. Math. CAM, 16, 280-305 (2011) · Zbl 1300.65076
[33] Abdulle, A., Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales, Math. Comput., 81, 278, 687-713 (2012) · Zbl 1243.65134
[34] Abdulle, A.; Schwab, C., Heterogeneous multiscale fem for diffusion problems on rough surfaces, SIAM Multiscale Model. Simul., 3, 1, 195-220 (2005) · Zbl 1160.65337
[35] Barrault, M.; Maday, Y.; Nguyen, N.; Patera, A., An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, C.R. Acad. Sci. Paris Ser. I, 339, 667-672 (2004) · Zbl 1061.65118
[36] Maday, Y.; Nguyen, N. C.; Patera, A. T.; Pau, G., A general multipurpose interpolation procedure: the magic points, Commun. Pure Appl. Anal., 8, 1, 383-404 (2009) · Zbl 1184.65020
[37] Binev, P.; Cohen, A.; Dahmen, W.; Devore, R.; Petrova, G.; Wojtaszczyk, P., Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43, 1457-1472 (2011) · Zbl 1229.65193
[38] Buffa, A.; Maday, Y.; Patera, A. T.; Prud’homme, C.; Turinici, G., A priori convergence of the greedy algorithm for the parametrized reduced basis, ESAIM-Math. Model. Numer. Anal., 46, 3, 595-603 (2012), (special issue in honor of David Gottlieb). · Zbl 1272.65084
[40] Zienkiewicz, O., The Finite Element Method (2005), Elsevier/Butterworth-Heinemann: Elsevier/Butterworth-Heinemann Amsterdam · Zbl 1084.74001
[41] Schmid, H. J., On cubature formulae with a minimal number of knots, Numer. Math., 31, 3, 281-297 (1978) · Zbl 0427.65014
[42] Clément, P., Approximation by finite element functions using local regularization, Rev. Fran aise Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique, 9, R2, 77-84 (1975) · Zbl 0368.65008
[44] Prudhomme, S.; Oden, J., On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Engrg., 176, 1-4, 313-331 (1999) · Zbl 0945.65123
[45] Schmidt, A.; Siebert, K., Design of adaptative finite element software: the finite element toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, vol. 42 (2005), Springer-Verlag: Springer-Verlag Berlin · Zbl 1068.65138
[46] Bangerth, W.; Rannacher, R., Adaptive Finite Element Methods for Differential Equations (2003), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1020.65058
[47] Nochetto, R.; Veeser, A.; Verani, M., A safeguarded dual weighted residual method, IMA J. Numer. Anal., 29, 1, 126-140 (2009) · Zbl 1168.65070
[49] Abdulle, A.; Nonnenmacher, A., A short and versatile finite element multiscale code for homogenization problem, Comput. Methods Appl. Mech. Engrg., 198, 37-40, 2839-2859 (2009) · Zbl 1229.74125
[50] Chen, Z.; Deng, W.; Ye, H., Upscaling of a class of nonlinear parabolic equations for the flow transport in heterogeneous porous media, Commun. Math. Sci., 3, 4, 493-515 (2005) · Zbl 1090.76058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.