scholarly journals Fortin operator for the Taylor–Hood element

2021 ◽  
Author(s):  
L. Diening ◽  
J. Storn ◽  
T. Tscherpel
Keyword(s):  
Finite Element ◽  
Bubble Functions ◽  
Element Pair

AbstractWe design a local Fortin operator for the lowest-order Taylor–Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $$P_2$$ P 2 –$$P_0$$ P 0 and the augmented Taylor–Hood element in 3D.

CONTINUOUS Q1–Q1 STOKES ELEMENTS STABILIZED WITH NON-CONFORMING NULL EDGE AVERAGE VELOCITY FUNCTIONS

2007 ◽  
Vol 17 (03) ◽  
pp. 439-459 ◽  
Author(s):  
LEOPOLDO P. FRANCA ◽  
SAULO P. OLIVEIRA ◽  
MARCUS SARKIS
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Stokes Equations ◽  
Least Squares Method ◽  
Approximation Properties ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Bubble Functions ◽  
Velocity Pressure ◽  
Element Method

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

Schwarz Type Solvers for -FEM Discretizations of Mixed Problems

2012 ◽  
Vol 12 (4) ◽  
pp. 369-390
Author(s):  
Sven Beuchler ◽  
Martin Purrucker
Keyword(s):  
Finite Element ◽  
Stokes Problem ◽  
Schur Complement ◽  
Algebraic Equations ◽  
The Finite Element Method ◽  
Contraction Ratio ◽  
Linear Algebraic Equations ◽  
Stokes Problems ◽  
Elasticity Problems ◽  
Element Pair

AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

A POSTERIORI ERROR ESTIMATORS VIA BUBBLE FUNCTIONS

1996 ◽  
Vol 06 (01) ◽  
pp. 33-41 ◽  
Author(s):  
ALESSANDRO RUSSO
Keyword(s):  
Finite Element ◽  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error Estimators ◽  
Element Space ◽  
A Posteriori Error ◽  
Error Estimators ◽  
Bubble Functions ◽  
Posteriori Error Estimators

In this paper we discuss a way to recover a classical residual-based error estimator for elliptic problems by using a finite element space enriched with bubble functions. The advection-dominated case is also discussed.

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