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.

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.

scholarly journals Characteristics Weak Galerkin Finite Element Methods for Convection-Dominated Diffusion Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ailing Zhu ◽  
Qiang Xu ◽  
Ziwen Jiang
Keyword(s):  
Finite Element ◽  
Optimal Order ◽  
Weak Galerkin ◽  
Order Error ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Galerkin Finite Element Methods ◽  
Finite Element Procedure ◽  
Convection Dominated ◽  
Diffusion Problems

The weak Galerkin finite element method is combined with the method of characteristics to treat the convection-diffusion problems on the triangular mesh. The optimal order error estimates inH1andL2norms are derived for the corresponding characteristics weak Galerkin finite element procedure. Numerical tests are performed and reported.

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 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.

scholarly journals Mixed methods for degenerate elliptic problems and application to fractional Laplacian

2020 ◽  
Author(s):  
María Eugenia Cejas ◽  
Ricardo Durán ◽  
Mariana Prieto
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Mixed Finite Element ◽  
Optimal Order ◽  
Order Error ◽  
Quadrilateral Elements ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Weighted Norms ◽  
Solutions Of Equations

  We analyze the approximation by mixed finite element methods of solutions of     equations of the form div  [[EQUATION]]  , where the coefficient a=a(x) can     degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the     coefficient $a$ belongs to the Muckenhoupt class  [[EQUATION]] .     The analysis developed applies to general mixed finite element spaces satisfying the     standard commutative diagram property, whenever some stability and interpolation     error estimates are valid in weighted norms. Next, we consider in detail the case     of Raviart-Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes.          For the lowest order case we show that the regularity assumprtion can be removed and prove  anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.

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