@TechReport{	  it:2010-026,
  author	= {Xin He and Maya Neytcheva and Stefano Serra Capizzano},
  title		= {On an Augmented {L}agrangian-Based Preconditioning of
		  {O}seen Type Problems},
  institution	= {Department of Information Technology, Uppsala University},
  department	= {Division of Scientific Computing},
  year		= {2010},
  number	= {2010-026},
  month		= nov,
  abstract	= {The paper deals with a general framework for constructing
		  preconditioners for saddle point matrices, in particular as
		  arising in the discrete linearized Navier-Stokes equations
		  (Oseen's problem). We utilize the so-called augmented
		  Lagrangian approach, where the original linear system of
		  equations is first transformed to an equivalent one, which
		  latter is then solved by a preconditioned iterative
		  solution method.
		  
		  The matrices in the linear systems, arising after the
		  discretization of Oseen's problem, are of two-by-two block
		  form as are the best known preconditioners for these. In
		  the augmented Lagrangian formulation, a scalar
		  regularization parameter is involved, which strongly
		  influences the quality of the block-preconditioners for the
		  system matrix (referred to as 'outer'), as well as the
		  conditioning and the solution of systems with the resulting
		  pivot block (referred to as 'inner') which, in the case of
		  large scale numerical simulations has also to be solved
		  using an iterative method. We analyse the impact of the
		  value of the regularization parameter on the convergence of
		  both outer and inner solution methods.
		  
		  The particular preconditioner used in this work exploits
		  the inverse of the pressure mass matrix. We study the
		  effect of various approximations of that inverse on the
		  performance of the preconditioners, in particular that of a
		  sparse approximate inverse, computed in an
		  element-by-element fashion. We analyse and compare the
		  spectra of the preconditioned matrices for the different
		  approximations and show that the resulting preconditioner
		  is independent of problem, discretization and method
		  parameters, namely, viscosity, mesh size, mesh anisotropy.
		  
		  We also discuss possible approaches to solve the modified
		  pivot matrix block.}
}