The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decomposethe computational domain into a series of subdomains that are deformationsof a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation,but solved for different choices of deformations of the referencesubdomains and mapped onto the reference shape.The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mappedfrom the generic computational part to the actual computational part.We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition,to a posteriori error estimation, and to “gluing” the local solutions together in the multidomain case.

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space – and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.

We present a technique for the rapid and reliable prediction oflinear-functionaloutputs of elliptic coercive partial differential equations with affineparameter dependence. The essential components are (i )(provably) rapidlyconvergent global reduced-basis approximations – Galerkin projectiononto a spaceW N spanned by solutions of the governing partial differentialequation at Nselected points in parameter space; (ii ) a posteriori error estimation– relaxations of the error-residual equation that provideinexpensive bounds for the error in the outputs of interest; and (iii ) off-line/on-line computational procedures – methods whichdecouple the generationand projection stages of the approximation process. The operationcount for theon-line stage – in which, given a new parameter value, we calculatethe output ofinterest and associated error bound – depends only on N (typicallyvery small) andthe parametric complexity of the problem; the method is thus ideallysuited for therepeated and rapid evaluations required in the context of parameter estimation,design, optimization, and real-time control. In our earlier work we develop a rigorous a posteriori error bound framework for reduced-basisapproximations of elliptic coercive equations. The resulting errorestimates are, in some cases, quite sharp: the ratio of the estimatederror in the output to the true error in the output, or effectivity , is close to (but always greater than) unity. However, inother cases, the necessary “bound conditioners” – in essence,operator preconditioners that (i ) satisfy an additional spectral“bound” requirement, and (ii ) admit the reduced-basisoff-line/on-line computational stratagem – either can not be found, oryield unacceptably large effectivities. In this paper we introduce a newclass of improved bound conditioners: the critical innovation is thedirect approximation of the parametric dependence of theinverse of the operator (rather than the operator itself); wethereby accommodate higher-order (e.g., piecewise linear) effectivityconstructions while simultaneously preserving on-line efficiency. Simpleconvex analysis and elementary approximation theory suffice to prove thenecessary bounding and convergence properties.