Numerical study of Navier-Stokes problems with generalized Robin boundary conditions
The first goal of this project is to study a reduced order model for the Fluid-Structure Interaction (FSI) problem. We present the assumptions made to achieve the reduced order formulation and its fully discretized numerical scheme. One of the characteristic of the model is that the fluid equation is endowed with a non standard boundary condition called Generalized Robin Boundary Condition (GRBC) which requires some extra regularity of the solution. We study the well-posedness of a Laplacian problem with GRBC and the convergence of the finite element scheme. We numerically solve the latter problem and compare the convergence order obtained with the theoretical results. Then we study the SUPG (streamline/ upwind Petrov/Galerkin) and PSPG (pressure-stabilizing/Petrov-Galerkin) stabilization schemes for incompressible flows. We stabilize the flow equation of the reduced order model and compare the simulation obtained with the use of the finite elements pairs P1-bubble/P1 and P1/P1 with stabilization. Finally, we derive the formulation on a reference domain of the boundary term for the reduced order model; moreover, by introducing a symmetric affine transformation we write explicitly its affine decomposition. For all the computation we use the C++ library LifeV; the meshes used are generated by means of the Gmsh library.