A Static condensation Reduced Basis Element method: approximation and a posteriori error estimation

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 eigen- function "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, com- putational performance, and also the effectivity of the a posteriori error bounds.

[1]  W. Hurty.  On the dynamic analysis of structural systems using component modes , 1964 .

[2]  B. Haasdonk,et al.  A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems , 2011 .

[3]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[4]  Gianluigi Rozza,et al.  A reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks , 2012 .

[5]  John W. Peterson,et al.  A high-performance parallel implementation of the certified reduced basis method , 2011 .

[6]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[7]  J. Hesthaven,et al.  A reduced basis method for multiple electromagnetic scattering in three dimensions , 2011 .

[8]  O. Widlund,et al.  Iterative Methods for the Solution of Elliptic Problems on Regions, Partitioned Into Substructures , 2015 .

[9]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[10]  Yvon Maday,et al.  The Reduced Basis Element Method: Application to a Thermal Fin Problem , 2004, SIAM J. Sci. Comput..

[11]  A. Bermúdez,et al.  Galerkin lumped parameter methods for transient problems , 2011 .

[12]  B. Häggblad,et al.  Model reduction methods for dynamic analyses of large structures , 1993 .

[13]  Yvon Maday,et al.  A Reduced-Basis Element Method , 2002, J. Sci. Comput..

[14]  U. Hetmaniuk,et al.  A SPECIAL FINITE ELEMENT METHOD BASED ON COMPONENT MODE SYNTHESIS , 2010 .

[15]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[16]  P. Seshu,et al.  Substructuring and Component Mode Synthesis , 1997 .

[17]  Harbir Antil,et al.  Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system , 2011, Optim. Methods Softw..

[18]  Harbir Antil,et al.  Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables , 2010, Comput. Vis. Sci..

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .

[21]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[22]  Anthony T. Patera,et al.  An "hp" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations , 2010, SIAM J. Sci. Comput..

[23]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[24]  F. Bourquin,et al.  Component mode synthesis and eigenvalues of second order operators : discretization and algorithm , 1992 .

[25]  Jan S. Hesthaven,et al.  A seamless reduced basis element method for 2D Maxwell"s problem: An introduction , 2011, CSE 2011.

[26]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

[27]  Anthony T. Patera,et al.  A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations , 2002, J. Sci. Comput..

[28]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[29]  Richard B. Lehoucq,et al.  An Automated Multilevel Substructuring Method for Eigenspace Computation in Linear Elastodynamics , 2004, SIAM J. Sci. Comput..

[30]  Ngoc Cuong Nguyen,et al.  A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales , 2008, J. Comput. Phys..

[31]  JL Eftang,et al.  Adaptive Port Reduction in Static Condensation , 2012 .

[32]  S. Sen.  Reduced-Basis Approximation and A Posteriori Error Estimation for Many-Parameter Heat Conduction Problems , 2008 .

[33]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[34]  Mats G. Larson,et al.  Adaptive component mode synthesis in linear elasticity , 2011 .

[35]  Susanne C. Brenner,et al.  The condition number of the Schur complement in domain decomposition , 1999, Numerische Mathematik.