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

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 WNWN spanned by solutions of the governing equation at N selected points in parameter space (chosen by an adaptive procedure to minimize the estimated error and the effectivity; (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 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 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. 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.