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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 OfflineOnline computational decomposition strategies -minimum marginal cost for high performance in the realtime/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.

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Abstract.In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems involving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numerical results are presented to assess our approach.Mathematics Subject Classification. 35J25, 35J60, 35K15, 35K55.