共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
在Poisson方程的求解域Ω存在一致的三角剖分,并且相邻两初始单元构成平行四边形的假设下,证明了若Poisson方程的解u属于H6(Ω),那么二次有限元的误差有h4的渐近展开.基于误差的渐近展开,可以利用h4-Richardson外推进一步提高数值解的精度阶,并且能够得到一个后验误差估计.最后,一个数值算例验证了理论分析. 相似文献
3.
In this article, we propose a new finite element space Λ$_h$ for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space Λ$_h$ is designed in such a way that the strong requirement V$_h\subset$Λ$_h$ in [9] is weakened to {v$_h\in$V$_h$; divv$_h$=0}$\subset$Λ$_h$ so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new Λ$_h$ coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d, d=2,3$ and for triangular partitions in $\mathbb{R}^2$. Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings. 相似文献
4.
Lie-Heng Wang 《计算数学(英文版)》1985,3(3):211-218
In this paper, the isoparametric element of 2-degree Lagrange type for second order elliptic P.D.E, with nonhomogeneous Dirichlet boundary value is considered. We prove $\|u-u_h\|_{1,\Omega}=O(h^2)$, which is the same as $\|u-u_h\|_{1,\Omega_h}=O(h^2)$ where $\Omega$ and $\Omega_h$ in $R^2$ are the domain of the boundary value problem and isoparametric triangulation domain respectively. 相似文献
5.
Jun-Bin Gao & T. M. Shih 《计算数学(英文版)》1994,12(3):248-258
1.IntroductionThesolutionoftheC'-continuityrequlrementofKirchhoffbendingwithfiniteele-mentmodelsresult8incomplicatedhigher.l...nt.I2J'[4'I7].Besidesthelargenumberofunknowns,difficultiesmayalsoarisefrommiredsecondderiVativesattheverticestakenasnodalvari.bl.I8l.Toovercomesuchdifficulties,asplittingsplineelemelltmethodisintrod.cedl5j'l9],butthisalwayscausescomplicatedcomputation-nomthepracticalpoilitofviewlower-degreepolynomialfiniteelemelitsaremoredesirable.Unfortunately,thesimpleelementsbase… 相似文献
6.
The Alternating-direction Implicit Characteristic Finite Element Methods for the Three-dimensional Generalized Nerve Conduction Equation
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Zhiyue Zhang 《偏微分方程(英文版)》2001,14(3):207-219
A complete convergence nnalysis is given for two-level scheme of the alternating-direction implicit characteristic finite element method for the approximate solution of the three-dimensional generalized nerve conduction equation. By use of the calculation of vector product, H^{-1} norm estimates, and a priori estimate theory and technique, the optimal order estimate in L² is obtained. 相似文献
7.
王玉玉 《数学年刊A辑(中文版)》2018,39(3):273-286
本文中,通过几何方法证明了σ相关同伦元素在球面稳定同伦群π_mS中是非平凡的,其中m=p~(n+1)q+2p~nq+(s+3)p~2q+(s+3)pq+(s+3)q-8,p≥7是奇素数,n3,0≤sp-3,且q=2(p-1).该σ相关同伦元素在Adams谱序列的E_2-项中由■_s+3■_ng0表示. 相似文献
8.
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
9.
Exponential Time Differencing-Padé Finite Element Method for Nonlinear Convection-Diffusion-Reaction Equations with Time Constant Delay
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In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional $L^2$ numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of $O(k^{3}+h^{r})$ (ETD3-Padé) or $O(k^{4}+h^{r})$ (ETD4-Padé) in the $L^2$ norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes. 相似文献
10.
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
11.
Let p be an odd prime. The authors detect a nontrivial element ã p of order p2 in the stable homotopy groups of spheres by the classical Adams spectral sequence. It is represented by \(a_0^{p - 2} h_1 \in Ext_A^{p - 1,pq + p - 2} (\mathbb{Z}/p,\mathbb{Z}/p)\) in the E2-term of the ASS and meanwhile p · ã p is the first periodic element α p . 相似文献
12.
Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation,
applied to isoparametric d-linear quadrilateral finite element approximation of functionals of multi scale solutions to second order elliptic partial
differential equations in bounded domains of ℝd. In contrast to the usual aim to derive an a posteriori error estimate, this work derives, as the mesh size tends to zero,
a uniformly convergent error expansion for the error density, with computable leading order term. It is shown that the optimal
adaptive isotropic mesh uses a number of elements proportional to the d/2 power of the
quasi-norm of the error density; the same error for approximation with a uniform mesh requires a number of elements proportional
to the d/2 power of the larger L1 norm of the same error density. A point is that this measure recognizes different convergence rates for multi scale problems,
although the convergence order may be the same. The main result is a proof that the adaptive algorithm based on successive
subdivisions of elements reduces the maximal error indicator with a factor or stops with the error asymptotically bounded
by the tolerance using the optimal number of elements, up to a problem independent factor. An important step is to prove uniform
convergence of the expansion for the error density, which is based on localized averages of second order difference quotients
of the primal and dual finite element solutions. The averages are used since the difference quotients themselves do not converge
pointwise for adapted meshes. The proof uses weak convergence techniques with a symmetrizer for the second order difference
quotients and a splitting of the error into a dominating contribution, from elements with no hanging nodes or edges on the
initial mesh, and a remaining asymptotically negligible part. Numerical experiments for an elasticity problem with a crack
and different variants of the averages show that the algorithm is useful in practice also for relatively large tolerances,
much larger than the small tolerances needed to theoretically guarantee that the algorithm works well.
AMS subject classification (2000) 65N12, 65N30, 65N50 相似文献
13.
In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-norm are presented for two-grid scheme, where $t,$ $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm. 相似文献
14.
Kai Wang & Na Wang 《计算数学(英文版)》2022,40(5):777-793
This article concerns numerical approximation of a parabolic interface problem with general $L^2$ initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the $k$-step backward difference formula with $ k=1,\ldots,6 $. To maintain high-order convergence in time for possibly nonsmooth $L^2$ initial value, we modify the standard backward difference formula at the first $k-1$ time levels by using a method recently developed for fractional evolution equations. An error bound of $\mathcal{O}(t_n^{-k}\tau^k+t_n^{-1}h^2|\log h|)$ is established for the fully discrete finite element method for general $L^2$ initial data. 相似文献
15.
有限域上互反本原正规元的存在性 总被引:1,自引:0,他引:1
设q是素数方幂,n是正整数,Fqn是qn个元素的有限域.本文证明了:当正整数n≥32时,对任意的素数方幂q,存在Fqn中的本原元ξ满足ξ和ξ-1都是Fqn 在Fq上的正规元,也即{ξ,ξq,…,ξqn-1}和{ξ-1,ξ-q,…,ξ-qn-1)都构成Fqn在Fq 上的本原正规基. 相似文献
16.
This paper develops a framework to deal with the unconditional superclose analysis of
nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example,
a new mixed finite element method (FEM) is established and the $τ$ -independent superclose
results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an
error splitting technique, with which the time-discrete and the spatial-discrete systems are
constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require
certain time step conditions. Finally, some numerical results are provided to confirm the
theoretical analysis, and show the efficiency of the proposed method. 相似文献
17.
Kang Li & Zhijun Tan 《高等学校计算数学学报(英文版)》2020,13(4):1050-1067
A two-grid finite element approximation is studied in the fully discrete
scheme obtained by discretizing in both space and time for a nonlinear hyperbolic
equation. The main idea of two-grid methods is to use a coarse-grid space ($S_H$) to
produce a rough approximation for the solution of nonlinear hyperbolic problems
and then use it as the initial guess on the fine-grid space ($S_h$). Error estimates are
presented in $H^1$-norm, which show that two-grid methods can achieve the optimal
convergence order as long as the two different girds satisfy $h$ = $\mathcal{O}$($H^2$). With the
proposed techniques, we can obtain the same accuracy as standard finite element
methods, and also save lots of time in calculation. Theoretical analyses and numerical examples are presented to confirm the methods. 相似文献
18.
Jun Hu 《计算数学(英文版)》2015,33(3):283-296
The design of mixed finite element methods in linear elasticity with symmetric stress approximations has been a longstanding open problem until Arnold and Winther designed the first family of mixed finite elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+1}$ tensors whose divergence is a $P_{k-1}$ polynomial on each triangle for $k$ ≥ 2. Such a two dimensional family was extended, by Arnold, Awanou and Winther, to a three dimensional family of mixed elements where the discrete stress space is the space of $H(div, Ω\;\mathbb{S}) — P_{k+2}$ tensors, whose divergence is a $P_{k-1}$ polynomial on each tetrahedron for $k$ ≥ 2. In this paper, we are able to construct, in a unified fashion, mixed finite element methods with symmetric stress approximations on an arbitrary simplex in $\mathbb{R}^n$ for any space dimension. On the contrary, the discrete stress space here is the space of $H(div, Ω\;\mathbb{S}) — P_k$ tensors, and the discrete displacement space here is the space of $L²(Ω ; \mathbb{R}^n) — P_{k-1}$ vectors for $k ≥ n$+1. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and can be regarded as extensions to any dimension of those in two and three dimensions by Hu and Zhang. 相似文献
19.
设 $p\geq 7$ 为任意奇素数. 证明了当 $3\leq s
相似文献
20.
讨论了拟线性对流占优扩散问题的数值模拟.对对流部分采用特征线格式进行离散,以消除流动锋线前沿的数值弥散现象,保证格式的稳定性;而对扩散部分采用扩展混合有限元方法,同时逼近未知函数,未知函数的梯度及伴随向量函数.理论分析和数值算例表明,此方法是稳定的,具有最优L2逼近精度. 相似文献