Numerical Analysis of Mixed Formulations for Bingham Fluids

Visco-plastic materials have been attracting a great amount of attention among researchers in the study of fluid flow due to their widespread presence in various fields of science. While numerical techniques for simulating their flow have seen significant improvements in the last few decades, the efficient solution of the nonlinear partial differential equations modelling them still poses many challenges. From a mathematical point of view, the major difficulty associated with the numerical solution of these equations is the presence of singularities in (a priori unknown) parts of the domain. This "irregularity" of the equations generally reflects in slow convergence of numerical solvers. In this thesis we introduce an augmented formulation of the Bingham visco-plastic flow which is aimed at circumventing the singularity of the equations. We develop a nonlinear solver based on this new formulation and compare its performance to other common techniques for solving the Bingham flow, indicating superior convergence properties of the solver based on the new formulation. Upon linearization and discretization of the augmented formulation, a sequence of linear systems is obtained which are in general very large and sparse. We introduce a nonlinear geometric multilevel technique for the efficient solution of these linear systems. The convergence of this multilevel technique is accelerated by a exible Krylov subspace method. We test the resulting numerical scheme on both academic test cases and problems arising from real-life applications with a particular emphasis on problems in hemodynamics.