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- Michele Benzi, Maxim A. Olshanskii
- SIAM J. Scientific Computing
- 2006

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)

- Maxim A. Olshanskii
- Numerical Lin. Alg. with Applic.
- 1999

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)

SUMMARY We study different variants of the augmented Lagrangian-based block triangular preconditioner introduced by the first two authors in [SIAM J. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied to various finite element and MAC discretizations of the Oseen problem in two and three… (More)

- Piotr P. Grinevich, Maxim A. Olshanskii
- SIAM J. Scientific Computing
- 2009

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)

- Maxim A. Olshanskii, Arnold Reusken, Jörg Grande
- SIAM J. Numerical Analysis
- 2009

AGalerkin finite element method is considered to approximate the incompressible Navier–Stokes equations together with iterative methods to solve a resulting system of algebraic equations. This system couples velocity and pressure unknowns, thus requiring a special technique for handling. We consider the Navier–Stokes equations in velocity–– kinematic… (More)

In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which… (More)

- Maxim A. Olshanskii, Arnold Reusken
- Math. Comput.
- 2004

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)

In this paper the grad–div stabilization for the incompressible Navier–Stokes finite element approximations is considered from two different viewpoints: (i) as a variational multiscale approach for the pressure subgrid modeling and (ii) as a stabilization procedure of least-square type. Some new error estimates for the linearized problem with the grad–div… (More)

- Maxim A. Olshanskii, Yuri V. Vassilevski
- SIAM J. Scientific Computing
- 2007

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)