Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer- ...

CutFEM: Discretizing geometry and partial differential equations

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Nonlinear Functional Analysis And Its Applications

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

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Finite element methods for surface PDEs

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

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An Augmented Lagrangian-Based Approach to the Oseen Problem

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 $H^1$- and in the $L^2$-norm. Results of numerical experiments are included that confirm this optimality.

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A Finite Element Method for Elliptic Equations on Surfaces

https://www.math.uh.edu/~molshan/ftp/pub/GradDivVMS.pdf

Grad–div stabilization and subgrid pressure models for the incompressible Navier–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.

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Grad-div stablilization for Stokes equations

http://www.mathcs.emory.edu/~molshan/ftp/pub/GLOS.pdf

Maxim A. Olshanskii

Papers

An Augmented Lagrangian-Based Approach to the Oseen Problem

A Finite Element Method for Elliptic Equations on Surfaces

Grad–div stabilization and subgrid pressure models for the incompressible Navier–Stokes equations

Grad-div stablilization for Stokes equations

Stabilized finite element schemes with LBB-stable elements for incompressible flows