共查询到20条相似文献,搜索用时 15 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.
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
3.
《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
4.
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
5.
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
6.
7.
We design and numerically validate a recovery based linear finite element method for solving the biharmonic equation.The main idea is to replace the gradient operator▽on linear finite element space by G(▽)in the weak formulation of the biharmonic equation,where G is the recovery operator which recovers the piecewise constant function into the linear finite element space.By operator G,Laplace operator△is replaced by▽·G(▽).Furthermore,the boundary condition on normal derivative▽u-n is treated by the boundary penalty method.The explicit matrix expression of the proposed method is also introduced.Numerical examples on the uniform and adaptive meshes are presented to illustrate the correctness and effectiveness of the proposed method. 相似文献
8.
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
9.
In this paper, we study the Crank-Nicolson Galerkin finite element method and construct a two-grid algorithm for the general two-dimensional time-dependent Schrödinger equation. Firstly, we analyze the superconvergence error estimate of the finite element solution in $H^1$ norm by use of the elliptic projection operator. Secondly, we propose a fully discrete two-grid finite element algorithm with Crank-Nicolson scheme in time. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two Poisson equations on the fine grid. Finally, we also derive error estimates of the two-grid finite element solution with the exact solution in $H^1$ norm. It is shown that the solution of two-grid algorithm can achieve asymptotically optimal accuracy as long as mesh sizes satisfy $H = \mathcal{O}(h^{\frac{1}{2}})$. 相似文献
10.
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
11.
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
12.
This article establishes a discrete maximum principle (DMP) for the approximate solution of convection–diffusion–reaction problems obtained from the weak Galerkin (WG) finite element method on nonuniform rectangular partitions. The DMP analysis is based on a simplified formulation of the WG involving only the approximating functions defined on the boundary of each element. The simplified weak Galerkin (SWG) method has a reduced computational complexity over the usual WG, and indeed provides a discretization scheme different from the WG when the reaction terms are present. An application of the SWG on uniform rectangular partitions yields some 5- and 7-point finite difference schemes for the second order elliptic equation. Numerical experiments are presented to verify the DMP and the accuracy of the scheme, particularly the finite difference scheme. 相似文献
13.
In this paper, a new discontinuous Galerkin method is developed for
the parabolic equation with jump coefficients satisfying the
continuous flow condition. Theoretical analysis shows that this
method is $L^2$ stable. When the finite element space consists of
interpolative polynomials of degrees $k$, the convergent rate of the
semi-discrete discontinuous Galerkin scheme has an order of$\mathcal{O}(h^k)$. Numerical examples for both 1-dimensional and
2-dimensional problems demonstrate the validity of the new method. 相似文献
14.
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
15.
Gwanghyun Jo & Do Y. Kwak 《高等学校计算数学学报(英文版)》2020,13(2):281-295
We introduce a low order finite element method for three dimensional elasticity problems. We extend Kouhia-Stenberg element [12] by using two nonconforming components and one conforming component, adding stabilizing terms to the associated bilinear form to ensure the discrete Korn's inequality. Using the second Strang's lemma, we show that our scheme has optimal convergence rates in $L^2$ and piecewise $H^1$-norms even when Poisson ratio $\nu$ approaches $1/2$. Even though some efforts have been made to design a low order method for three dimensional problems in [11,16], their method uses some higher degree basis functions. Our scheme is the first true low order method. We provide three numerical examples which support our analysis. We compute two examples having analytic solutions. We observe the optimal $L^2$ and $H^1$ errors for many different choices of Poisson ratios including the nearly incompressible cases. In the last example, we simulate the driven cavity problem. Our scheme shows non-locking phenomena for the driven cavity problems also. 相似文献
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17.
A global superconvergent $L^{\infty}$-error estimate of mixed finite element methods for semilinear elliptic optimal control problems 下载免费PDF全文
Li Li 《Journal of Applied Analysis & Computation》2015,5(3):313-328
In this paper, we discuss the superconvergence of mixed finite element methods for a semilinear elliptic control problem with an integral constraint. The state and costate are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximation of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that this approximation has convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results. 相似文献
18.
Khaled Omrani 《Numerical Methods for Partial Differential Equations》2004,20(4):527-551
In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 相似文献
19.
A superconvergent $L^{\infty}$-error estimate of RT1 mixed methods for elliptic control problems with an integral constraint 下载免费PDF全文
In this paper, we investigate the superconvergence property of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. A superconvergent approximation of the control variable $u$ will be constructed by a projection of the discrete adjoint state. It is proved that this approximation have convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results. 相似文献
20.
Alan Demlow. 《Mathematics of Computation》2004,73(248):1623-1653
In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-Douglas-Marini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.