弱有限元方法在线弹性问题中的应用

本文考虑弱有限元（简称WG）方法在线弹性问题中的应用.WG方法是传统有限元方法的推广,用于偏微分方程的数值求解.和传统有限元一样,它的基本思想源于变分原理.WG方法的特点是使用在剖分单元内部和剖分单元边界上分别有定义的分片多项式函数（即弱函数）作为近似函数来逼近真解,并针对弱函数定义相应的弱微分算子代入数值格式进行计算.除此之外,WG方法允许在数值格式中引进稳定子以实现近似函数的弱连续性.WG方法具有允许使用任意多边形或多面体剖分,数值格式与逼近函数构造简单,易于满足相应的稳定性条件等优点.本文考虑WG方法在求解线弹性问题中的应用.围绕线弹性问题数值求解中常见的三个问题,即：数值格式的强制性,闭锁性,应力张量的对称性介绍WG方法在线弹性问题求解中的应用.     

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H² norm is established for the corresponding WG finite element solutions. Error estimates in the usual L² norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014     

Convergence of Weak Galerkin Finite Element Method for Second Order Linear Wave Equation in Heterogeneous Media

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.     

A weak Galerkin finite element method for the stokes equations

This paper introduces a weak Galerkin (WG) finite element method for the Stokes equations in the primal velocity-pressure formulation. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k≥1 for the velocity and polynomials of degree k?1 for the pressure, both are discontinuous. The velocity element is enhanced by polynomials of degree k?1 on the interface of the finite element partition. All the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly-defined spaces. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. It must be emphasized that the WG finite element method is designed on finite element partitions consisting of arbitrary shape of polygons or polyhedra which are shape regular.     

随机平面线弹性问题的一类弱Galerkin方法

本文针对带有随机杨氏模量和荷载的平面线弹性问题,提出了一类随机弱Galerkin有限元方法.先利用Karhunen-Loève展开把随机项参数化,将方程转化为一个确定性问题;再采用弱Galerkin有限元法和$k$-/$p$-型方法分别离散空间区域和随机场.在弱Galerkin离散中,用分片$s（s\geqslant 1$）和$s+1$次多项式逼近单元内部的应力和位移,用分片$s$次多项式逼近位移在单元边界上的迹.证明了该方法关于空间网格尺度最优且与Lamé常数$\lambda$一致无关的误差估计.最后通过数值算例验证了理论结果.     

弱有限元方法简论

本文简述弱有限元方法(weak Galerkin finite element met,hods)的数学基本原理和计算机实现.弱有限元方法对间断函数引入广义弱微分,并将其应用于偏微分方程相应的变分形式进行数值求解,而数值解的弱连续性则通过稳定子或光滑子来实现.弱有限元方法针对广义函数而构建,是经典有限元方法的一种自然拓广,且能够弥补经典有限元方法的某些缺憾,也因此在科学与工程计算领域具有广泛的应用前景.     

Numerical and theoretical study of weak Galerkin finite element solutions of Turing patterns in reaction–diffusion systems

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P₀, P₀, RT₀) . The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme.     

Weak Galerkin finite element method for a class of quasilinear elliptic problems

The purpose of this paper is to study the weak Galerkin finite element method for a class of quasilinear elliptic problems. The weak Galerkin finite element scheme is proved to have a unique solution with the assumption that guarantees the corresponding operator to be strongly monotone and Lipschitz-continuous. An optimal error estimate in a mesh-dependent energy norm is established. Some numerical results are presented to confirm the theoretical analysis.     

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

In this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L² norm is established. Numerical experiments are presented to validate the theoretical analysis.     

求解Brinkman方程的修正弱有限元方法

本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性.     

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two‐point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Constructing Order Two Superconvergent WG Finite Elements on Rectangular Meshes

In this paper, we introduce a stabilizer free weak Galerkin (SFWG) finite element method for second order elliptic problems on rectangular meshes. With a special weak Gradient space, an order two superconvergence for the SFWG finite element solution is obtained, in both $L^2$ and $H^1$ norms. A local post-process lifts such a $P_k$ weak Galerkin solution to an optimal order $P_{k+2}$ solution. The numerical results confirm the theory.     

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

This paper presents error estimates in both an energy norm and the $L^2$-norm for the weak Galerkin (WG) finite element methods for elliptic problems with low regularity solutions. The error analysis for the continuous Galerkin finite element remains same regardless of regularity. A totally different analysis is needed for discontinuous finite element methods if the elliptic regularity is lower than H-1.5. Numerical results confirm the theoretical analysis.     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

A hybridized weak Galerkin finite element scheme for the Stokes equations

In this paper a hybridized weak Galerkin(HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced.The WG method uses weak functions and their weak derivatives which are defined as distributions.Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees.Different combination of polynomial spaces leads to different WG finite element methods,which makes WG methods highly flexible and efficient in practical computation.A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution.With this new feature,the HWG method can be used to deal with jumps of the functions and their flux easily.Optimal order error estimates are established for the corresponding HWG finite element approximations for both primal variables and the Lagrange multiplier.A Schur complement formulation of the HWG method is derived for implementation purpose.The validity of the theoretical results is demonstrated in numerical tests.     

A weak Galerkin method for second order elliptic problems with polynomial reduction

The second order elliptic equation, which is also know as the diffusion-convection equation, is of great interest in many branches of physics and industry. In this paper, we use the weak Galerkin finite element method to study the general second order elliptic equation. A weak Galerkin finite element method is proposed and analyzed. This scheme features piecewise polynomials of degree $k\geq 1$ on each element and piecewise polynomials of degree $k-1\geq 0$ on each edge or face of the element. Error estimates of optimal order of convergence rate are established in both discrete $H^1$ and standard $L^2$ norm. The paper also presents some numerical experiments to verify the efficiency of the method.     

A unified study of continuous and discontinuous Galerkin methods

A unified study is presented in this paper for the design and analysis of different finite element methods(FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs, discontinuous Galerkin(DG) methods, hybrid discontinuous Galerkin(HDG) methods and weak Galerkin(WG) methods.Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore,a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.     

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

In this paper, the weak Galerkin finite element method (WG-FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG-FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise-in-time error estimates in L²-norm and H¹-norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived.     

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM ₂ three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.