scholarly journals Numerical Analysis of anH1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Jinfeng Wang ◽  
Meng Zhao ◽  
Min Zhang ◽  
Yang Liu ◽  
Hong Li
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Finite Difference Methods ◽  
Mixed Finite Element Method ◽  
Telegraph Equation ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Spatial Direction ◽  
Coupled Equations ◽  
Fractional Telegraph Equation

We discuss and analyze anH1-Galerkin mixed finite element (H1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate anH1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying theH1-GMFE method. Based on the discussion on the theoretical error analysis inL2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown inH1-norm. Moreover, we derive and analyze the stability ofH1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.

The Numerical Analysis and Simulation of a Linearized Crank-Nicolson H1-GMFEM for Nonlinear Coupled BBM Equations

2014 ◽  
Vol 513-517 ◽  
pp. 1919-1926 ◽  
Author(s):  
Min Zhang ◽  
Zu Deng Yu ◽  
Yang Liu ◽  
Hong Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Scheme ◽  
Mixed Finite Element ◽  
A Priori ◽  
Numerical Test ◽  
Mixed Finite Element Method ◽  
Spatial Direction ◽  
Time Direction ◽  
Crank Nicolson

In this article, the numerical scheme of a linearized Crank-Nicolson (C-N) method based on H1-Galerkin mixed finite element method (H1-GMFEM) is studied and analyzed for nonlinear coupled BBM equations. In this method, the spatial direction is approximated by an H1-GMFEM and the time direction is discretized by a linearized Crank-Nicolson method. Some optimal a priori error results are derived for four important variables. For conforming the theoretical analysis, a numerical test is presented.

scholarly journals Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chenguang Zhou ◽  
Yongkui Zou ◽  
Shimin Chai ◽  
Fengshan Zhang
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Polynomial Functions ◽  
Weak Galerkin ◽  
Convergence Estimates

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

scholarly journals A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Dongyang Shi ◽  
Qili Tang ◽  
Yadong Zhang
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Auxiliary Variable ◽  
Diffusion Problem ◽  
Optimal Order ◽  
Order Error ◽  
Elliptic Projection ◽  
Indispensable Tool ◽  
Convection Dominated

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableuand the auxiliary variableσwith respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

scholarly journals Expanded Mixed Finite Element Method for the Two-Dimensional Sobolev Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Qing-li Zhao ◽  
Zong-cheng Li ◽  
You-zheng Ding
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Two Dimensional ◽  
Sobolev Equation ◽  
Order Error ◽  
Expanded Mixed Finite Element ◽  
Element Method

Expanded mixed finite element method is introduced to approximate the two-dimensional Sobolev equation. This formulation expands the standard mixed formulation in the sense that three unknown variables are explicitly treated. Existence and uniqueness of the numerical solution are demonstrated. Optimal order error estimates for both the scalar and two vector functions are established.

scholarly journals A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems in ℝ3

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Gwanghyun Jo ◽  
J. H. Kim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Order Convergence ◽  
New Family ◽  
Optimal Order Convergence

In this paper, we suggest a new patch condition for nonconforming mixed finite elements (MFEs) on parallelepiped and provide a framework for the convergence. Also, we introduce a new family of nonconforming MFE space satisfying the new patch condition. The numerical experiments show that the new MFE shows optimal order convergence in Hdiv and L2-norm for various problems with discontinuous coefficient case.

Error Estimates of H1-Galerkin Expanded Mixed Finite Element Methods for Heat Problems

2011 ◽  
Vol 267 ◽  
pp. 493-498
Author(s):  
Hai Tao Che ◽  
Mei Xia Li ◽  
Li Juan Liu
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Low Permeability ◽  
Heat Equations ◽  
Order Error ◽  
Mixed Element ◽  
Expanded Mixed Finite Element ◽  
Element Method

H1-Galerkin expanded mixed element method are discussed for a class of second-order heat equations. The methods possesses the advantage of mixed finite element while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. H1-Galerkin expanded mixed finite element method for heat equations are described, an optimal order error estimate for the methods is obtained.

H1 -Galerkin Expanded Mixed Finite Element Methods for Pseudo-Hyperbolicequations

2011 ◽  
Vol 268-270 ◽  
pp. 908-912 ◽  
Author(s):  
Mei Xia Li
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Order Error ◽  
Expanded Mixed Finite Element ◽  
Physical Quantities ◽  
Element Method

H1-Galerkin mixed finite element method combining with expanded mixed element method are discussed for a class of second-order pseudo-hyperbolic equations. The methods possesses the advantage of mixed finite element while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. Depended on the physical quantities of interest, the methods are discussed. The existence and uniqueness of numerical solutions of the scheme are derived and an optimal order error estimate for the methods is obtained.

ON THE CONVERGENCE OF A TRIANGULAR MIXED FINITE ELEMENT METHOD FOR REISSNER–MINDLIN PLATES

1996 ◽  
Vol 06 (03) ◽  
pp. 339-352 ◽  
Author(s):  
RICARDO G. DURÁN ◽  
ELSA LIBERMAN
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Mixed Finite Element ◽  
Plate Thickness ◽  
Mixed Finite Element Method ◽  
Mindlin Plate ◽  
Optimal Order ◽  
The Difference ◽  
Element Method

We analyze the convergence of a mixed finite element method introduced by Zienkiewicz, Taylor, Papadopoulos and Oñate for the Reissner–Mindlin plate model. In order to do this, we compare it with a method which is known to be convergent with optimal order uniformly in the plate thickness. We show that the difference between the solutions of both methods is of higher order than the error. In particular the method does not present locking and is optimal order convergent. We also present several numerical experiments which confirm the similar behavior of both methods.

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