共查询到20条相似文献,搜索用时 31 毫秒
1.
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. 相似文献
2.
Xiu Ye & Shangyou Zhang 《高等学校计算数学学报(英文版)》2023,16(1):230-241
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. 相似文献
3.
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. 相似文献
4.
5.
Leila Jafarian Khaled‐Abad Rezvan Salehi 《Numerical Methods for Partial Differential Equations》2021,37(1):302-340
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. 相似文献
6.
We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method. 相似文献
7.
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. 相似文献
8.
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. 相似文献
9.
This paper considers weak Galerkin finite element approximations on
polygonal/polyhedral meshes for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k (k ≥ 1)$ for the stress
approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We
show the existence and uniqueness of the semi-discrete and fully discrete solutions,
and derive optimal a priori error estimates. Numerical examples are provided to
support the theoretical analysis. 相似文献
10.
A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions 下载免费PDF全文
Saqib Hussain Nolisa Malluwawadu Peng Zhu 《Journal of Applied Analysis & Computation》2018,8(5):1452-1463
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. 相似文献
11.
《Numerical Methods for Partial Differential Equations》2018,34(3):1009-1032
In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H1 and L2 norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory. 相似文献
12.
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. 相似文献
13.
崔明 《数学物理学报(A辑)》2001,21(3):364-372
考虑裂缝 孔隙介质中地下水污染问题均匀化模型的数值模拟.对压力方程采用混合元方法,对浓度方程采用Galerkin交替方向有限元方法,对吸附浓度方程采用标准Galerkin方法,证明了交替方向有限元格式具有最优犔2 和犎1 模误差估计. 相似文献
14.
Rui Li Jian Li Xin Liu Zhangxin Chen 《Numerical Methods for Partial Differential Equations》2017,33(4):1352-1373
In this article, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity‐pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers‐Joseph‐Saffman law. By using the weak Galerkin approach, we consider the two‐dimensional problem with the piecewise constant elements for approximations of the velocity, pressure, and hydraulic head. Stability and optimal error estimates are obtained. Finally, we provide several numerical results illustrating the good performance of the proposed scheme and confirming the optimal order of convergence provided by the weak Galerkin approximation. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1352–1373, 2017 相似文献
15.
Weak Galerkin finite element methods combined with Crank-Nicolson scheme for parabolic interface problems 下载免费PDF全文
This article is devoted to the a priori error estimates of the fully discrete Crank-Nicolson approximation for the linear parabolic interface problem via weak Galerkin finite element methods (WG-FEM). All the finite element functions are discontinuous for which the usual gradient operator is implemented as distributions in properly defined spaces. Optimal order error estimates in both $L^{\infty}(H^1)$ and $L^{\infty}(L^2)$ norms are established for lowest order WG finite element space $({\cal P}_{k}(K),\;{\cal P}_{k-1}(\partial K),\;\big[{\cal P}_{k-1}(K)\big]^2)$. Finally, we give numerical examples to verify the theoretical results. 相似文献
16.
In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method. 相似文献
17.
Lin Mu Junping Wang Xiu Ye 《Numerical Methods for Partial Differential Equations》2014,30(3):1003-1029
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 相似文献
18.
Shimin Chai Yongkui Zou Chenguang Zhou Wenju Zhao 《Numerical Methods for Partial Differential Equations》2019,35(5):1745-1755
This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L2 and an H2 type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results. 相似文献
19.
New numerical techniques are presented for the solution of the
two-dimensional time fractional evolution equation in the unit
square. In these methods, Galerkin finite element is used for the
spatial discretization, and, for the time stepping, new alternating
direction implicit (ADI) method based on the backward Euler method
combined with the first order convolution quadrature approximating
the integral term are considered. The ADI Galerkin finite element
method is proved to be convergent in time and in the $L^2$ norm in
space. The convergence order is$\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is
the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to
support our theoretical analysis. 相似文献
20.
Qiaoluan H. Li Junping Wang 《Numerical Methods for Partial Differential Equations》2013,29(6):2004-2024
A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H1 and L2 norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献