scholarly journals SUPG Approximation for the Oseen Viscoelastic Fluid Flow with Stabilized Lowest-Equal Order Mixed Finite Element Method

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 128
Author(s):  
Shahid Hussain ◽  
Afshan Batool ◽  
Md. Al Mahbub ◽  
Nasrin Nasu ◽  
Jiaping Yu
Keyword(s):  
Finite Element ◽  
Fluid Flow ◽  
Viscoelastic Fluid ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Constitutive Law ◽  
Optimal Error Estimates ◽  
Fe Method ◽  
The Stability ◽  
Stability And Accuracy

In this article, a stabilized mixed finite element (FE) method for the Oseen viscoelastic fluid flow (OVFF) obeying an Oldroyd-B type constitutive law is proposed and investigated by using the Streamline Upwind Petrov–Galerkin (SUPG) method. To find the approximate solution of velocity, pressure and stress tensor, we choose lowest-equal order FE triples P 1 - P 1 - P 1 , respectively. However, it is well known that these elements do not fulfill the i n f - s u p condition. Due to the violation of the main stability condition for mixed FE method, the system becomes unstable. To overcome this difficulty, a standard stabilization term is added in finite element variational formulation. The technique is applied herein possesses attractive features, such as parameter-free, flexible in computation and does not require any higher-order derivatives. The stability analysis and optimal error estimates are obtained. Three benchmark numerical tests are carried out to assess the stability and accuracy of the stabilized lowest-equal order feature of the OVFF.

scholarly journals Theoretical stability analysis of mixed finite element model of shale-gas flow with geomechanical effect

2020 ◽  
Vol 75 ◽  
pp. 33
Author(s):  
Mohamed F. El-Amin ◽  
Jisheng Kou ◽  
Shuyu Sun
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Shale Gas ◽  
Gas Flow ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Element Model ◽  
Mathematical Proofs ◽  
The Stability ◽  
Shale Gas Transport

In this work, we introduce a theoretical foundation of the stability analysis of the mixed finite element solution to the problem of shale-gas transport in fractured porous media with geomechanical effects. The differential system was solved numerically by the Mixed Finite Element Method (MFEM). The results include seven lemmas and a theorem with rigorous mathematical proofs. The stability analysis presents the boundedness condition of the MFE solution.

Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations

2020 ◽  
Author(s):  
Huadong Gao ◽  
Weiwei Sun ◽  
Chengda Wu
Keyword(s):  
Finite Element ◽  
Potential Field ◽  
Order Approximation ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Second Order ◽  
Optimal Error Estimates ◽  
Order Accuracy ◽  
Recovery Technique ◽  
Electric Potential Field

Abstract This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperature in such a strongly coupled system, while previous work only showed the first-order accuracy $O(h)$ for all three components in a traditional way. In this paper, we prove that the method produces the optimal second-order accuracy $O(h^2)$ for $u$ in the spatial direction, although the accuracy for the potential/field is in the order of $O(h)$. And more importantly, we propose a simple one-step recovery technique to obtain a new numerical electric potential/field of second-order accuracy. The analysis presented in this paper relies on an $H^{-1}$-norm estimate of the mixed finite element methods and analysis on a nonclassical elliptic map. We provide numerical experiments in both two- and three-dimensional spaces to confirm our theoretical analyses.

scholarly journals An Optimal Error Estimates ofH1-Galerkin Expanded Mixed Finite Element Methods for Nonlinear Viscoelasticity-Type Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Haitao Che ◽  
Yiju Wang ◽  
Zhaojie Zhou
Keyword(s):  
Finite Element ◽  
Nonlinear Viscoelasticity ◽  
Mixed Finite Element ◽  
A Priori ◽  
Type Equation ◽  
Mixed Finite Element Method ◽  
Optimal Error Estimates ◽  
Uniqueness Of Solutions ◽  
Expanded Mixed Finite Element ◽  
Element Method

We investigate aH1-Galerkin mixed finite element method for nonlinear viscoelasticity equations based onH1-Galerkin method and expanded mixed element method. The existence and uniqueness of solutions to the numerical scheme are proved. A priori error estimation is derived for the unknown function, the gradient function, and the flux.

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