A Reduced Basis Element Method for Complex Flow Systems

Abstract

The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations within domains belonging to a certain class. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into smaller blocks that are topologically similar to a few reference shapes (or generic computational parts). Associated with each reference shape are precomputed solutions corresponding to the same governing partial differential equation, and similar boundary conditions, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to the computational domain is then taken to be a linear combination of the precomputed solutions, mapped from the reference shapes for the different blocks to the actual domain. The variation of the geometry induces non-affine parameter dependence, and we apply the empirical interpolation technique to achieve an offline/online decoupling of the reduced basis procedure. Some results for incompressible flow systems have already been presented, and the focus here will be to further improve the offline/online decoupling of problems with non-affine parameter dependence. To this end we use the empirical interpolation method to approximate the parameter depen- dent operators. We also present a generalized transfinite interpolation method intended to produce global C1 mappings from the reference shapes to each corresponding block of the computational domain.