Reduced Basis Modeling of 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. The basic idea is to f irst decompose the computational domain into a series of subdomains that are deformations of a few reference domains. Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but s olved for different choices of deformations of the reference subdomains and mapped onto the reference geometry. The approximation associated with a new geometry is then taken to be a linear combination of the precomputed solutions, mapped from the referen ce domain for the part to the actual domain. In this project, we propose to extend earlier work on stationary flows to solve time-dependent flows in hierarchical flow systems with moving boundaries, and with application to biological flows. The researc h will focus on the following aspects: (i) reduced basis methods for time-dependent problems; (ii) reduced basis modeling of fluid-structure interaction; (iii) extension of the reduced basis method to three space dimensions; (iv) certification of the s imulation results via a posteriori error analysis; (v) optimal selection of precomputed solutions. The research will be done in close collaboration with Prof. Einar M. Rønquist at NTNU, and two research groups involved in the modeling and simulation o f blood flow: one at Laboratoire Jacques Louis Lions at Universite Pierre et Marie Curie in Paris (Prof. Yvon Maday) and one at Simula Research Laboratory in Oslo (Dr. Ola Skavhaug).