A New Weak Galerkin Finite Element Scheme for the Brinkman Model

2016 ◽  
Vol 19 (5) ◽  
pp. 1409-1434 ◽  
Author(s):  
Qilong Zhai ◽  
Ran Zhang ◽  
Lin Mu
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Optimal Order ◽  
Brinkman Model ◽  
Weak Galerkin ◽  
Order Error ◽  
Brinkman Equations ◽  
Generalized Distributions ◽  
Variational Form ◽  
Weak Derivatives

AbstractThe Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

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 Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

2021 ◽  
Author(s):  
şuayip toprakseven ◽  
Peng Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Reaction ◽  
Stability And Convergence ◽  
Optimal Order ◽  
Galerkin Finite Element Method ◽  
Weak Galerkin ◽  
Order Error ◽  
Galerkin Finite Element ◽  
Element Method

In this paper, a modified weak Galerkin finite element method on Shishkin mesh has been developed and analyzed for the singularly perturbed convection-diffusion-reaction problems. The proposed method is based on the idea of replacing the standard gradient (derivative) and convection derivative by modified weak gradient (derivative) and modified weak convection derivative, respectively, over piecewise polynomials of degree $k\geq1$. The present method is parameter-free and has less degree of freedom compared to the weak Galerkin finite element method. Stability and convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy norm are proved. The method is uniformly convergent, i.e., the results hold uniformly regardless of the value of the perturbation parameter. Numerical experiments confirm these theoretical findings on Shishkin meshes. The numerical examples are also carried out on B-S meshes to confirm the theoretical results. Moreover, the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2-$ norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L_\infty-$ norm.

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 Analysis of finite element methods for vector Laplacians on surfaces

2019 ◽  
Vol 40 (3) ◽  
pp. 1652-1701 ◽  
Author(s):  
Peter Hansbo ◽  
Mats G Larson ◽  
Karl Larsson
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Vector Fields ◽  
Optimal Order ◽  
Order Error ◽  
Optimal Order Error Estimates ◽  
Elastic Membranes ◽  
Tangential Vector ◽  
Higher Order Finite Elements ◽  
Lagrange Elements

Abstract We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in ${\mathbb{R}}^3$. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a ${\mathbb{R}}^3$ vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and $L^2$ norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.

scholarly journals Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model

1997 ◽  
Vol 07 (02) ◽  
pp. 217-238 ◽  
Author(s):  
Douglas N. Arnold ◽  
Richard S. Falk
Keyword(s):  
Finite Element ◽  
Plate Thickness ◽  
Mesh Size ◽  
Mindlin Plate ◽  
Optimal Order ◽  
Plate Model ◽  
Order Error ◽  
Linear Finite Element ◽  
Optimal Order Error Estimates ◽  
Linear Elements

An analysis is presented for a recently proposed finite element method for the Reissner–Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations confirm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t fixed, the method does not converge as the mesh size h tends to zero.

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 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.

scholarly journals Hybridized weak Galerkin finite element methods for Brinkman equations

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Jiwei Jia ◽  
◽  
Young-Ju Lee ◽  
Yue Feng ◽  
Zichan Wang ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Weak Galerkin ◽  
Galerkin Finite Element ◽  
Brinkman Equations ◽  
Galerkin Finite Element Methods

A C 0 conforming dg finite element method for biharmonic equations on triangle/tetrahedron

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiu Ye ◽  
Shangyou Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Optimal Order ◽  
Biharmonic Equations ◽  
Order Error ◽  
Simple Formulation ◽  
Piecewise Polynomials ◽  
Second Gradient ◽  
Strong Gradient ◽  
Element Method

Abstract A C 0 conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C 0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C 1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence.

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