Reduced Basis Methods for Elliptic Equations in sub-domains with A-Posteriori Error Bounds and Adaptivity

MOX 27

Reduced Basis Methods for Elliptic Equations in sub-domains with A-Posteriori Error Bounds and Adaptivity

Thursday 2nd October 2003

Rozza, Gianluigi

We present an application in multi-parametrized subdomains based on a technique for the rapid and reliable prediction of linear-functional output of elliptic coercive partial differential equations with affine parameter dependence (reduced basis methods). The essential components are (i) (provably) rapidly convergent global reduce-basis approximation - Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space (ii) a posteriori 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 wich 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 N (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. In [11] a rigorous a posteriori error bound framework for reduced-basis approximations of elliptic coercive equations is developed. The resulting error estimates are, in some cases, quite sharp: the ratio of the estimated error in the output to the true error in the output, or effectivity, is close to (but always greater than) unity. We use a posteriori bound error estimator applied also to an adaptive procedure in choosing the approximation space and its dimension, minimizing the estimated erro of the effectivity [23]. The application is based on a parametrized geometry, divided in subdomains, each of them depending by geometrical quantities that can be useful for future haemodynamics applications [15], such as the bypass configuration problem (stenosis lenght, graft angle, artery diameter, incoming bypass diameter and outflow lenght). For future development guidelines we suggest to see [19] and [16].