Analysis of estimators for Stokes problem using a mixed approximation

El Akkad Abdeslam,Ahmed Elkhalfi

doi:10.5269/bspm.41354

Abstract

In this work, we introduce the steady Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of Stokes equations. It includes algorithms for discretization by mixed finite element methods. We use a block diagonal preconditioners for Stokes problem. We obtain a faster convergence when applying the preconditioned MINRES method. Two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true discretization error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

Highlights

A wealth of literature on solving saddle point systems exists, much of it related to particular applications
The finite element subspaces Xh and M h are constructed in the usual manner so that the inclusion Xh × M h ⊂ V × W holds
This bound can be pessimistic, it will still provide some insight into the effect of α on preconditioned MINRES convergence

Summary

Model problem

We will consider the model of viscous incompressible flow in an idealized, bounded, connected domain in R2,. The boundary value problem which is posed on two dimensional domains Ω, is defined as: Cβ : β −→u + (∇−→u − pI)−→n = −→g in Γ =: ∂Ω. Vector is the field −→u is divergence tohpeerflautiodr,vtehloeciftuyn, cptioisnatlhe−→f prinesstuhree field, space [L2(Ω)]2, −→g in the space [L2(Γ)]2, the pressure p in the space L2(Ω) and β is a nonzero bounded continuous function defined on ∂Ω. If β is strictly positive constant such that β ≻≻ 1 Cβ , is the Dirichlet boundary condition and if β ≺≺ 1 the Cβ, is the Neumann boundary condition. Let the bilinear forms a : V × V −→ R, b : V × W −→ R and d : W × W −→ R a(−→u , −→v ) = ∇−→u .∇−→v dx + β −→u .−→v , Γ (2.6).

Finite element approximation

Block diagonal preconditioners

Analysis of estimators

Numerical simulation

Findings

Conclusion