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We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique… (More)

We present a rigorous numerical analysis and computational tests for the Galerkin finite element discretization of the velocity-vorticity-helicity formulation of the equilibrium Navier-Stokes equations (NSE). This formulation was recently derived by the authors, is the first NSE formulation that directly solves for helicity, the first velocity-vorticity… (More)

The paper concerns with an iterative technique for solving discretized Stokes type equations with varying viscosity coefficient. We build a special block preconditioner for the discrete system of equations and perform an analysis revealing its properties. The subject of this paper is motivated by numerical solution of incompressible non-Newtonian fluid… (More)

In this paper a stabilizing augmented Lagrangian technique for the Stokes equations is studied. The method is consistent and hence does not change the continuous solution. We show that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient. We analyze the influence of this stabilization on the… (More)

For the three-dimensional incompressible Navier-Stokes equations, we present a formulation featuring velocity, vorticity and helical density as independent variables. We find the helical density can be observed as a Lagrange multiplier corresponding to the divergence-free constraint on the vorticity variable, similar to the pressure in the case of the… (More)

We consider several preconditioners for the pressure Schur complement of the discrete steady Oseen problem. Two of the preconditioners are well known from the literature and the other is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem, these approaches give rise to a family of the block preconditioners for the… (More)

Incompressible unsteady Navier–Stokes equations in pressure – velocity variables are considered. By use of the implicit and semi-implicit schemes presented the resulting system of linear equations can be solved by a robust and efficient iterative method. This iterative solver is constructed for the system of linearized Navier–Stokes equations. The Schur… (More)

We consider an abstract parameter dependent saddle-point problem and present a general framework for analyzing robust Schur complement preconditioners. The abstract analysis is applied to a generalized Stokes problem, which yields robustness of the Cahouet-Chabard preconditioner. Motivated by models for two-phase incompressible flows we consider a… (More)

In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in R d defines a d-dimensional space-time manifold in the space-time continuum R d+1. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational… (More)