Fast Iterative Methods for The Incompressible Navier-Stokes Equations

Content Provider	Semantic Scholar
Author	Rehman, Mujeeb Ur
Copyright Year	2010
Abstract	This thesis has been completed in partial fulfillment of the requirements of Delft University of Technology (Delft, The Netherlands) for the award of the Ph.D. degree. Cover: A numerical solution of 2D driven cavity flow Stokes problem on 132 × 132 Q2-Q1 FEM grid. Summary Efficient numerical solution of the incompressible Navier-Stokes equations is a hot topic of research in the scientific computing community. In this thesis efficient linear solvers for these equations are developed. The finite element discretization of the incompressible Navier-Stokes equations gives rise to a nonlinear system. This system is linearized with Picard or Newton type methods. Due to the incompressibility equation the resulting linear equations are of saddle point type. Saddle point problems also occur in many other engineering fields. They pose extra problems for the solvers and therefore efficient solution of such systems of equations forms an important research activity. In this thesis we discuss preconditioned Krylov methods, that are developed for saddle point problems. The most direct and easy applicable strategy to solve linear system of equations arising from Navier-Stokes is to apply preconditioners of ILU-type. This type of pre-conditioners is based on the coefficients of the matrix but not on knowledge of the system. In general, without precautions, they fail for saddle point problems. To overcome this problem, pivoting or renumbering of nodal points is necessary. Direct methods also suffer from the same problem, i.e zeros may arise at the main diagonal. Renum-bering is used to reduce the profile or bandwidth of the matrix. To avoid zero pivots it is necessary to use extra information of the discretized equations. First we start with a suitable node renumbering scheme like Sloan or Cuthill-McKee to get an optimal profile. Thereafter unknowns are reordered per level such that zero pivots move to the end of each level. In this way unknowns are intermixed and the matrix can be considered as a sequence of smaller subsystems. This provides a reasonable efficient preconditioner if combined with ILU. We call it Saddle point ILU (SILU). A completely different strategy is based on segregation of velocity and pressure. This is done by so-called block preconditioners. These preconditioners are all based on SIMPLE or Uzawa type schemes. The idea is to solve the coupled system with a Krylov method and to accelerate the convergence by the block preconditioners. The expensive steps in the preconditioning is the solution of the velocity and …
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