Some stabilized finite element methods for the Stokes problem are reviewed. The Douglas-Wang approach confirms better stability features for high order interpolations. Next, the advective-diffusive model is approximated in the light of various stabilized methods, a global convergence analysis is presented and numerical experiments are performed. Biquadratic elements produce better numerical results under all stabilized methods examined. The design of the stability parameter is confirmed to be a crucial ingredient for simulating the advective-diffusive model, and some improved possibilities are suggested. Combinations of these methodologies are given in the conclusions and will be examined in detail in the sequel to this paper applied to the incompressible Navier-Stokes equations.

https://hal.inria.fr/inria-00075259/document

Stabilized finite element methods. I: Application to the advective-diffusive model

A new finite element formulation for computational fluid dynamics. X - The compressible Euler and Navier-Stokes equations

This paper describes recent advances in stability analysis that combine the limit theorems of classical plasticity with finite elements to give rigorous upper and lower bounds on the failure load. These methods, known as finite-element limit analysis, do not require assumptions to be made about the mode of failure, and use only simple strength parameters that are familiar to geotechnical engineers. The bounding properties of the solutions are invaluable in practice, and enable accurate limit loads to be obtained through the use of an exact error estimate and automatic adaptive meshing procedures. The methods are very general, and can deal with heterogeneous soil profiles, anisotropic strength characteristics, fissured soils, discontinuities, complicated boundary conditions, and complex loading in both two and three dimensions. A new development, which incorporates pore water pressures in finite-element limit analysis, is also described. Following a brief outline of the new techniques, stability solutions ...

Geotechnical stability analysis

http://amasud.web.engr.illinois.edu/Papers/Stabilized_Methods-F-H-M-FEM-Book-2004.pdf

Stabilized Finite Element Methods

We outline the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier-Stokes or Euler equations. Full incompressibility can be dealt with if the algorithm is used in its semi-explicit corm and its structure permits arbitrary interpolation cunctions to be used avoiding the Babuska-Brezzi restriction. In a fully explicit version it introduces a rational form of balancing dissipation avoiding the use of arbitrary parameters and forms for this

A general algorithm for compressible and incompressible flow—Part I. the split, characteristic‐based scheme

A systematic procedure to obtain the appropriate upwind direction and associated Petrov-Galerkin weighting function for the advection-diffusion equation is presented. It is shown that sharp internal and boundary layers are accurately approximated with this new method.

A consistent approximate upwind Petrov—Galerkin method for convection-dominated problems

Adaptive finite element computational fluid dynamics using an anisotropic error estimator

Finite element analysis of convection dominated reaction-diffusion problems

An adaptive Petrov-Galerkin formulation for the compressible Euler and Navier-Stokes equations

In this paper the Petrov-Galerkin method introduced in Vol. 68, pp. 83–95 is adaptively applied to convection-dominated problems. To this end a feedback function is created which increases or decreases the control of the gradient of the approximate solution. This leads to a method with good stability properties close to boundary layers and high accuracy where regular solutions do occur.

Augusto Cesar Galeao

Papers

A consistent approximate upwind Petrov—Galerkin method for convection-dominated problems

Adaptive finite element computational fluid dynamics using an anisotropic error estimator

Finite element analysis of convection dominated reaction-diffusion problems

An adaptive Petrov-Galerkin formulation for the compressible Euler and Navier-Stokes equations

Feedback Petrov-Galerkin methods for convection-dominated problems