共查询到20条相似文献,搜索用时 15 毫秒
1.
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
2.
Huoyuan Duan & Roger C.E. Tan 《计算数学(英文版)》2020,38(2):254-290
This paper is devoted to the establishment of sharper $a$ $priori$stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity $\nu$ and a reaction constant $\sigma$. With the establishment of sharper stability estimates and the help of $ad$ $hoc$finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity $\nu$, the reaction constant $\sigma$, and the mesh size $h$. Our analysis reveals that the viscosity $\nu$ and the reaction constant $\sigma$ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity $\nu$ and a large reaction constant $\sigma$. The sharper error estimates agree very well with the numerical results. 相似文献
3.
In this paper, we consider the numerical solution of the flame front equation,
which is one of the most fundamental equations for modeling combustion theory.
A schema combining a finite difference approach in the time direction and a spectral
method for the space discretization is proposed. We give a detailed analysis for the
proposed schema by providing some stability and error estimates in a particular case.
For the general case, although we are unable to provide a rigorous proof for the stability,
some numerical experiments are carried out to verify the efficiency of the schema.
Our numerical results show that the stable solution manifolds have a simple structure
when $\beta$ is small, while they become more complex as the bifurcation parameter $\beta$ increases. At last numerical experiments were performed to support the claim the
solution of flame front equation preserves the same structure as K-S equation. 相似文献
4.
Superconvergence Analysis of a BDF-Galerkin FEM for the Nonlinear Klein-Gordon-Schrödinger Equations with Damping Mechanism 下载免费PDF全文
Dongyang Shi & Houchao Zhang 《计算数学(英文版)》2023,41(2):224-245
The focus of this paper is on a linearized backward differential formula (BDF) scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations (KGSEs) with damping mechanism. Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme. The proof consists of three ingredients. First, a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms. Second, optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms. Third, by virtue of the relationship between the Ritz projection and the interpolation, as well as a so-called "lifting'' technique, the superconvergence behavior of order $O(h^2+\tau^2)$ in $H^1$-norm for the original variables are deduced. Finally, a numerical experiment is conducted to confirm our theoretical analysis. Here, $h$ is the spatial subdivision parameter, and $\tau$ is the time step. 相似文献
5.
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
6.
Finite element algorithm based on high-order time approximation for time fractional convection-diffusion equation 下载免费PDF全文
Xinfei Liu Yang Liu Hong Li Zhichao Fang Jinfeng Wang 《Journal of Applied Analysis & Computation》2018,8(1):229-249
In this paper, finite element method with high-order approximation for time fractional derivative is considered and discussed to find the numerical solution of time fractional convection-diffusion equation. Some lemmas are introduced and proved, further the stability and error estimates are discussed and analyzed, respectively. The convergence result $O(h^{r+1}+\tau^{3-\alpha})$ can be derived, which illustrates that time convergence rate is higher than the order $(2-\alpha)$ derived by $L1$-approximation. Finally, to validate our theoretical results, some computing data are provided. 相似文献
7.
We propose and analyze a $C^0$-weak Galerkin (WG) finite element method for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, the system of Navier-Stokes equations is reduced to a single fourth-order nonlinear partial differential equation and the incompressibility constraint is automatically satisfied. The proposed method uses continuous piecewise-polynomial approximations of degree $k+2$ for the stream-function $\psi$ and discontinuous piecewise-polynomial approximations of degree $k+1$ for the trace of $\nabla\psi$ on the interelement boundaries. The existence of a discrete solution is proved by means of a topological degree argument, while the uniqueness is obtained under a data smallness condition. An optimal error estimate is obtained in $L^2$-norm, $H^1$-norm and broken $H^2$-norm. Numerical tests are presented to demonstrate the theoretical results. 相似文献
8.
In this paper, we derive and analyze a conservative Crank-Nicolson-type
finite difference scheme for the Klein-Gordon-Dirac (KGD) system. Differing from
the derivation of the existing numerical methods given in literature where the numerical schemes are proposed by directly discretizing the KGD system, we translate
the KGD equations into an equivalent system by introducing an auxiliary function,
then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the
equivalent system. The scheme perfectly inherits the mass and energy conservative properties possessed by the KGD, while the energy preserved by the existing
conservative numerical schemes expressed by two-level's solution at each time step.
By using energy method together with the 'cut-off' function technique, we establish
the optimal error estimate of the numerical solution, and the convergence rate is $\mathcal{O}(τ^2 + h^2)$ in $l^∞$-norm with time step $τ$ and mesh size $h.$ Numerical experiments
are carried out to support our theoretical conclusions. 相似文献
9.
In this paper, the theoretical and numerical determination of a solely time-dependent load distribution is investigated for a simply supported non-homogeneous Euler–Bernoulli beam. The missing source is recovered from an additional “local” integral measurement. The existence and uniqueness of a solution to the corresponding variational problem is proved by employing Rothe’s method. This method also reveals a time-discrete numerical scheme based on the backward Euler method to approximate the solution. Corresponding error estimates are proved and assessed by two numerical experiments. 相似文献
10.
Etienne Emmrich 《BIT Numerical Mathematics》2011,51(3):581-607
A class of discontinuous Galerkin methods is studied for the time discretisation of the initial-value problem for a nonlinear
first-order evolution equation that is governed by a monotone, coercive, and hemicontinuous operator. The numerical solution
is shown to converge towards the weak solution of the original problem. Furthermore, well-posedness of the time-discrete problem
as well as a priori error estimates for sufficiently smooth exact solutions are studied. 相似文献
11.
Jun Zhao. 《Mathematics of Computation》2004,73(247):1089-1105
We provide an error analysis of finite element methods for solving time-dependent Maxwell problem using Nedelec and Thomas-Raviart elements. We study the regularity of the solution and develop some new error estimates of Nedelec finite elements. As a result, the optimal -error bound for the semidiscrete scheme is obtained.
12.
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
13.
Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous $P_1$ vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and $L^2$-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of $L^2$-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis. 相似文献
14.
Yongyong Cai & Yan Wang 《数学研究》2020,53(2):125-142
We consider the nonlinear Dirac equation (NLD) with time dependent external electro-magnetic potentials, involving a dimensionless parameter $ε\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime $ε\ll1$ (speed of light tends to infinity), we decompose the solution into the eigenspaces associated with the 'free Dirac operator' and construct an approximation to the NLD with $O(ε^2)$ error. The NLD converges (with a phase factor) to a coupled nonlinear Schrödinger system (NLS) with external electric potential in the nonrelativistic limit as $ε\to0^+$, and the error of the NLS approximation is first order $O(ε)$. The constructed $O(ε^2)$ approximation is well-suited for numerical purposes. 相似文献
15.
O. Steinbach 《Numerische Mathematik》2014,126(1):173-197
In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in $\widetilde{H}^{1/2}(\Gamma )$ . In addition to error estimates in the energy norm we also provide, by applying the Aubin–Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including $L_2(\Gamma )$ . The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results. 相似文献
16.
Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations 下载免费PDF全文
In this paper, two multiscale time integrators (MTIs), motivated from two
types of multiscale decomposition by either frequency or frequency and amplitude, are
proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0 < \varepsilon≤ 1.$ In fact, the solution to this equation
propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon≪1,$ which brings significantly
numerical burdens in practical computation. We rigorously establish two independent
error bounds for the two MTIs at $O(\tau^2/\varepsilon^2)$ and $O(\varepsilon^2)$ for $\varepsilon ∈ (0,1]$ with $\tau > 0$ as step
size, which imply that the two MTIs converge uniformly with linear convergence rate
at $O(\tau)$ for $ε ∈ (0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the
regimes when either $ε=O(1)$ or $0<ε≤\tau.$ Thus the meshing strategy requirement (or $ε$-scalability) of the two MTIs is $\tau =O(1)$ for $0<ε≪1,$ which is significantly improved
from $\tau =O(ε^3)$ and $\tau =O(ε^2)$ requested by finite difference methods and exponential
wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards
better understanding on the convergence and resolution properties of the two MTIs.
In addition, numerical results support the two error bounds very well. 相似文献
17.
In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the -contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.
18.
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. 相似文献
19.
Sheng-Jiang Cheng 《计算数学(英文版)》1985,3(4):315-319
In this paper we use an iterative method to get an approximate solution $u^n$ and $\bar{u}^n$ which approximate the exact solution $u$ with the error estimates $\|u-u^n\|+ch\|u-u^n\|_1+\|u-\bar{u}^n\|_1\leq ch^{n+2}$. 相似文献
20.
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