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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)

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)

- 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)

- 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)

- Maxim A. Olshanskii, Arnold Reusken
- Computing
- 2000

- Maxim A. Olshanskii, Jörg Peters, Arnold Reusken
- Numerische Mathematik
- 2006

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)

- 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, Xianmin Xu
- SIAM J. Numerical Analysis
- 2014

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)

- 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)

Conforming finite-element approximations are considered for the incompressible Navier– Stokes equations with nonlinear terms written in the convection or rotation forms. Implicit time integration results in nice stability properties of auxiliary problems which can be solved by efficient numerical algorithms. The original nonlinear system admits relatively… (More)