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Abstract

In this thesis, we address coupled incompressible flow problems with respect to their efficient numerical solutions. These problems are modeled by the Oseen equations, the Navier-Stokes equations and the Brinkman equations. For numerical approximations of these equations, we discretize these systems by Hdiv-conforming discontinuous Galerkin method which globally satisfy the divergence free velocity constraint on discrete level. The algebraic systems arising from discretizations are large in size and have poor spectral properties which makes it challenging to solve these linear systems efficiently. For efficient solution of these algebraic system, we develop our solvers based on classical iterative solvers preconditioned with multigrid preconditioners employing overlapping Schwarz smoothers of multiplicative type. Multigrid methods are well known for their robustness in context of self-adjoint problems. We present an overview of the convergence analysis of multigrid method for symmetric problems. However, we extend this method to non self-adjoint problems, like the Oseen equations, by incorporating the downwind ordering schemes of Bey and Hackbusch and we show the robustness of this method by empirical results. Furthermore, we extend this approach to non-linear problems, like the Navier-Stokes and the non-linear Brinkman equations, by using a Picard iteration scheme for linearization. We investigate extensively by performing numerical experiment for various examples of incompressible flow problems and show by empirical results that the multigrid method is efficient and robust with respect to the mesh size, the Reynolds number and the polynomial degree. We also observe from our numerical results that in case of highly heterogeneous media, multigrid method is robust with respect to a high contrast in permeability.