A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

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 [8, 16, 17], and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to 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 some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work [14, 15] in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to constructing the basis functions, to the mapping of the velocity fields, to satisfying the inf-sup condition, and to “gluing” the local solutions together in the multidomain case [4]. We also demonstrate an algorithm for choosing the most ∗Corresponding author: [email protected] efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.