Maxim A. Olshanskii

An Augmented Lagrangian-Based Approach to the Oseen Problem

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 introduced by Schoberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2-P0 and isoP2-P1 finite elements in support of our conclusions. We also show results of a comparison with two state-of-the-art preconditioners, showing the competitiveness of our approach.

Grad-div stablilization for Stokes equations

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 accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.

A Finite Element Method for Elliptic Equations on Surfaces

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 there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the

A low order Galerkin finite element method for the Navier–Stokes equations of steady incompressible flow: a stabilization issue and iterative methods

H^1

Pressure Schur Complement Preconditioners for the Discrete Oseen Problem

- and in the

Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations

L^2

Analysis of a Stokes interface problem

-norm. Results of numerical experiments are included that confirm this optimality.

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

A Galerkin 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 pressure variables as well as in velocity––Bernoulli pressure variables. The latter leads to the rotation form of nonlinear terms. This form of the equations plays an important role in our studies. A consistent stabilization method is considered from a new view point. Theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied. � 2002 Published by Elsevier Science B.V.

A finite element method for surface PDEs: matrix properties

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 matrix of the discrete Oseen system. In the paper we critically review possible advantages and difficulties of using various Schur complement preconditioners. We recall existing eigenvalue bounds for the preconditioned Schur complement and prove such for the newly proposed preconditioner. These bounds hold both for LBB stable and stabilized finite elements. Results of numerical experiments for several model two-dimensional and three-dimensional problems are presented. In the experiments we use LBB stable finite element methods on uniform triangular and tetrahedral meshes. One particular conclusion is that in spite of essential improvement in comparison with “simple” scaled mass-matrix preconditioners in certain cases, none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex (e.g., circulating) advection vector field.

ERROR ANALYSIS OF A SPACE-TIME FINITE ELEMENT METHOD FOR SOLVING PDES ON EVOLVING SURFACES ∗

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 generalized Stokes interface problem. Application of the general theory results in a new Schur complement preconditioner for this class of problems. The robustness of this preconditioner with respect to several parameters is treated. Results of numerical experiments are given that illustrate robustness properties of the preconditioner.