Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations

G. Rozza (1) , D B P Huynh (2) , A. T Patera (3)

1 MIT-MECHE - Department of Mechanical Engineering [Massachusetts Institute of Technology]
2 NUS - National University of Singapore
3 MIT - Massachusetts Institute of Technology

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth " para-metric manifold " —dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence ; a posteriori error estimation procedures—rigorous and sharp bounds for the linear-functional outputs of interest ; and Offline-Online computational decomposition strategies—minimum marginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, con-This work was supported by DARPA/AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425, the Singapore-MIT Alliance, the Pappalardo MIT Mechanical Engineering Graduate Monograph Fund, and the Progetto Roberto Rocca Politecnico di Milano-MIT. We acknowledge many helpful discussions with Professor Yvon Maday and many-query (e.g., design optimization, multi-model/scale) contexts. We present illustrative results for heat conduction and convection-diffusion, inviscid flow, and linear elasticity; outputs include transport rates, added mass, and stress intensity factors.

Domaines: Mécanique des structures [physics.class-ph]

G. Rozza, D B P Huynh, A. T Patera. Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations: Application to Transport and Continuum Mechanics. Archives of Computational Methods in Engineering, 2008, 15 (3), pp.229 - 275. ⟨10.1007/s11831-008-9019-9⟩. ⟨hal-01722593⟩