Stabilized finite element formulations for flow problems

This work focuses on developing novel stabilized finite element formulations for flow problems suitable for monolithic approaches to fluid-structure interaction. In addition to the challenges produced by the poor performance of the classical Galerkin method for mixed finite elements, monolithic methods for nonlinear problems require consistent linearization. One of the main contributions of the work is a consistent linearization of a variational multiscale formulation for incompressible Navier-Stokes. We compare the convergence of this formulation with a fixed-point iteration technique. We show for a number of example problems that the Newton-Raphson technique converges quadratically. For transient problems, we show that consistent Newton-Raphson approach does not converge in the vicinity of bifurcations such as the onset of unsteady vortex shedding. We also present a Schur's complement implementation of the consistent method that may be considered an extension of static condensation to nonlinear problems. The Newton-Schur technique greatly increases parallel performance, which helps offset the computational cost produced by the decomposition of the multiscale framework. For a number of three-dimensional problems (at Reynolds number up to 1000) we show that the Newton-Schur approach is scalable for reasonable problem sizes. To investigate the central challenges to numerically modeling the Navier-Stokes equations, we begin with a study of the Stokes and advection-diffusion equations. From our study of the Stokes equations we learn that stabilized methods based on enriching the velocity trial function space with bubble functions are only stable for triangular elements. We also present a stabilized formulation for the advection-diffusion equation using the generalized finite element framework. This formulation eliminates oscillations in the neighborhood of sharp gradients and requires fewer degrees of freedom than variational multiscale methods.