Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity

We present an application in multi-parametrized sub-domains based on a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence (reduced-basis methods). The main components are (i) rapidly convergent global reduced-basis approximations-Galerkin projection onto a space <i>W</i><sub>N</sub> spanned by solutions of the governing equation at <i>N</i> selected points in parameter space (chosen by an adaptive procedure to minimize the estimated error and the <i>effectivity</i>; (ii) <i>a posteriori</i> error estimation-relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures-methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage-in which, given a new parameter value, we calculate the output of interest and associated error bound-depends only on <i>N</i> (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. The application is based on a heat transfer problem in a parametrized geometry in view of haemodynamics applications and biomechanical devices optimization, such as the bypass configuration problem. [All rights reserved Elsevier]


Published in:
Applied Numerical Mathematics, 55, 4, 403-24
Year:
2005
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