A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

2017 ◽  
Vol 10 (1) ◽  
pp. 22-43 ◽  
Author(s):  
Peiqi Huang ◽  
Zhilin Li
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Discontinuous Coefficients ◽  
Brinkman Equation ◽  
Optimal Error Estimates ◽  
Discrete Problem ◽  
Brinkman Equations ◽  
Approximation Errors ◽  
Interior Penalties ◽  
Uniformly Stable

AbstractA nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

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.

Lagrangian Mixed Finite Element Methods for Nonlinear Thin Shell Problems

2020 ◽  
Vol 20 (4) ◽  
pp. 631-642 ◽  
Author(s):  
Mikhail M. Karchevsky
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Elastic Shell ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Mixed Finite Element Methods ◽  
Systems Of Nonlinear Equations ◽  
Discrete Problem ◽  
Second Derivatives

AbstractA class of Lagrangian mixed finite element methods is constructed for an approximate solution of a problem of nonlinear thin elastic shell theory, namely, the problem of finding critical points of the functional of potential energy according to the Budiansky–Sanders model. The proposed numerical method is based on the use of the second derivatives of the deflection as auxiliary variables. Sufficient conditions for the solvability of the corresponding discrete problem are obtained. Accuracy estimates for approximate solutions are established. Iterative methods for solving the corresponding systems of nonlinear equations are proposed and investigated.

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 A Mixed Finite Element Formulation for the Conservative Fractional Diffusion Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Suxiang Yang ◽  
Huanzhen Chen
Keyword(s):  
Finite Element ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Singular Term ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Element Formulation ◽  
Optimal Error

We consider a boundary-value problem of one-side conservative elliptic equation involving Riemann-Liouville fractional integral. The appearance of the singular term in the solution leads to lower regularity of the solution of the equation, so to the lower order convergence rate for the numerical solution. In this paper, by the dividing of equation, we drop the lower regularity term in the solution successfully and get a new fractional elliptic equation which has full regularity. We present a theoretical framework of mixed finite element approximation to the new fractional elliptic equation and derive the error estimates for unknown function, its derivative, and fractional-order flux. Some numerical results are illustrated to confirm the optimal error estimates.

scholarly journals Low-Order Nonconforming Mixed Finite Element Methods for Stationary Incompressible Magnetohydrodynamics Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Dongyang Shi ◽  
Zhiyun Yu
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Approximate Solutions ◽  
Mixed Finite Element Methods ◽  
Optimal Error Estimates ◽  
Stationary Incompressible Magnetohydrodynamics ◽  
Friedrichs Inequality ◽  
Incompressible Magnetohydrodynamics ◽  
Low Order

The nonconforming mixed finite element methods (NMFEMs) are introduced and analyzed for the numerical discretization of a nonlinear, fully coupled stationary incompressible magnetohydrodynamics (MHD) problem in 3D. A family of the low-order elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field, and the magnetic field. The existence and uniqueness of the approximate solutions are shown, and the optimal error estimates for the corresponding unknown variables inL2-norm are established, as well as those in a brokenH1-norm for the velocity and the magnetic fields. Furthermore, a new approach is adopted to prove the discrete Poincaré-Friedrichs inequality, which is easier than that of the previous literature.

Error Estimates and Iterative Procedure for Mixed Finite Element Solution of Second-Order Quasi-Linear Elliptic Problems

2004 ◽  
Vol 4 (4) ◽  
pp. 445-463 ◽  
Author(s):  
Mikhail Karchevsky ◽  
Alexander Fedotov
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Elliptic Equations ◽  
Iterative Procedure ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Second Order ◽  
Discrete Problem ◽  
Arbitrary Power ◽  
Power Rate

AbstractThe mixed finite element method for second-order quasi-linear elliptic equations with nonlinearities of arbitrary power rate of growth is considered. Error estimates are obtained. An iterative method for corresponding discrete problem is proposed and investigated.

NUMERICAL METHODS OF NEW MIXED FINITE ELEMENT SCHEME FOR SINGLE-PHASE COMPRESSIBLE FLOW

2013 ◽  
Vol 11 (01) ◽  
pp. 1350055 ◽  
Author(s):  
SHUYING ZHAI ◽  
XINLONG FENG ◽  
ZHIFENG WENG
Keyword(s):  
Finite Element ◽  
Compressible Flow ◽  
Single Phase ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Conforming Finite Element

In this paper, a new mixed finite element scheme is given based on the less regularity of velocity for the single phase compressible flow in practice. Based on the new mixed variational formulation, we give its stable conforming finite element approximation for the P0–P1 pair and its stabilized conforming finite element approximation for the P1–P1 pair. Moreover, optimal error estimates are derived in H1-norm and L2-norm for the approximation of pressure and error estimate in L2-norm for the approximation of velocity by using two methods. Finally, numerical tests confirm the theoretical results of our methods.

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