## Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics

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 parametric 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, control) 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.

Published in:
Archives of Computational Methods in Engineering, 15, 3, 229-275
Year:
2008
Keywords:
Note:
Invited review paper on the occasion of the ECCOMAS PhD Award received by the first author in 2006.
Laboratories: