Nonconforming P1 Finite Element for the Biharmonic Problem and Its Application to Stokes Problem

2021 ◽  
Author(s):  
N. Staïli ◽  
M. Rhoudaf
Keyword(s):  
Finite Element ◽  
Error Estimate ◽  
Numerical Experiments ◽  
Stokes Problem ◽  
Two Dimensional ◽  
Driven Cavity ◽  
Function Formulation ◽  
Stream Functions ◽  
Stream Function Formulation ◽  
The Laplace Operator

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming [Formula: see text] finite element. For the solutions which admit a regularity greater than [Formula: see text], in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in [Formula: see text] an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.

scholarly journals Finite element error analysis of surface Stokes equations in stream function formulation

2020 ◽  
Vol 54 (6) ◽  
pp. 2069-2097
Author(s):  
Philip Brandner ◽  
Arnold Reusken
Keyword(s):  
Finite Element ◽  
Error Analysis ◽  
Stream Function ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Coupled System ◽  
Discretization Method ◽  
Element Discretization ◽  
Function Formulation ◽  
Stream Function Formulation

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface Γ ⊂ ℝ3 without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk–Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.

scholarly journals Polygonal Finite Element for Two-Dimensional Lid-Driven Cavity Flow

2022 ◽  
Vol 70 (3) ◽  
pp. 4217-4239
Author(s):  
T. Vu-Huu ◽  
C. Le-Thanh ◽  
H. Nguyen-Xuan ◽  
M. Abdel-Wahab
Keyword(s):  
Finite Element ◽  
Cavity Flow ◽  
Two Dimensional ◽  
Driven Cavity ◽  
Driven Cavity Flow ◽  
Polygonal Finite Element

A Finite Element Scheme for Incompressible Viscous Flow Calculations

1985 ◽  
Author(s):  
Wang Li Cheng ◽  
Zhang Hui Ming
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Pressure Field ◽  
Viscous Flows ◽  
Cavity Flow ◽  
Finite Element Scheme ◽  
Driven Cavity ◽  
Element Scheme ◽  
Numerical Result ◽  
Incompressible Viscous Flow

A finite element scheme for two dimensional incompressible viscous flows in primitive variables is proposed in this paper. An upwind factor finite element method is devised to solve the momentum equations, and the continuity equation is satisfied by the correction of the pressure field. Numerical experiments are carried out for a driven cavity and a diffuser. The Renolds Number for the cavity flow is 100.0, and for the diffuser is 50000.0. The numerical result of the scheme for the cavity flow is compared with that by another numerical method and satistactory agreement is found.

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