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1.
Haiyan Su Xinlong Feng Yinnian He 《Numerical Methods for Partial Differential Equations》2017,33(3):681-703
This survey enfolds rigorous analysis of the defect‐correction finite element (FE) method for the time‐dependent conduction‐convection problem which based on the Crank‐Nicolson scheme. The method consists of two steps: solve a nonlinear problem with an added artificial viscosity term on a FE grid and correct the solutions on the same grid using a linearized defect‐correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. As a consequence, the effectiveness of the method to deal with high Reynolds number is illustrated in several numerical experiments. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 681–703, 2017 相似文献
2.
Yinnian He 《Numerical Methods for Partial Differential Equations》2012,28(1):155-187
In this article, we study the stability and convergence of the Crank‐Nicolson/Adams‐Bashforth scheme for the two‐dimensional nonstationary Navier‐Stokes equations with a nonsmooth initial data. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the implicit Crank‐Nicolson scheme for the linear terms and the explicit Adams‐Bashforth scheme for the nonlinear term. Moreover, we prove that the scheme is almost unconditionally stable for a nonsmooth initial data u0 with div u0 = 0, i.e., the time step τ satisfies: τ ≤ C0 if u0 ∈ H1 ∩ L∞; τ |log h| ≤ C0 if u0 ∈ H1 for the mesh size h and some positive constant C0. Finally, we obtain some error estimates for the discrete velocity and pressure under the above stability condition. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 28: 155‐187, 2012 相似文献
3.
Yuhong Zhang Yanren Hou Li Shan 《Numerical Methods for Partial Differential Equations》2015,31(6):2169-2208
In this article, we consider the time‐discrete method for three‐dimensional incompressible magnetohydrodynamics (MHD) equations. The Crank–Nicolson extrapolation scheme is used for time discretization. From the previous articles, under the assumption that the solution has high regularity which cannot be realistically assumed, the convergence of this scheme is optimal two‐order. In this article, under modest assumptions of initial values and the body force, we prove some new regularity results of the MHD equations. In addition, we derive the unconditional convergence of our scheme, but the convergent order is not optimal. Furthermore, we provide another conditional convergence estimation to increase the order. It is shown that the convergent rate increase half order in ‐norm, and at least a quarter order increased in ‐norm than the uncondtional results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2169–2208, 2015 相似文献
4.
Tong Zhang JinYun Yuan ZhiYong Si 《Numerical Methods for Partial Differential Equations》2015,31(6):2135-2168
In this article, a decoupled two grid finite element method (FEM) is proposed and analyzed for the nonsteady natural convection problem using the coarse grid numerical solutions to decouple the nonlinear coupled terms, and the corresponding optimal error estimates are derived. Compared with the standard Galerkin FEM and the usual two‐grid FEM, our algorithm not only keeps good accuracy but also saves a lot of computational cost. Some numerical examples are provided to verify the performances of the decoupled two‐grid FEM. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the decoupled two‐grid FEM for the nonsteady natural convection problem. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2135–2168, 2015 相似文献
5.
We combine fourth‐order boundary value methods (BVMs) for discretizing the temporal variable with fourth‐order compact difference scheme for discretizing the spatial variable to solve one‐dimensional heat equations. This class of new compact difference schemes achieve fourth‐order accuracy in both temporal and spatial variables and are unconditionally stable due to the favorable stability property of BVMs. Numerical results are presented to demonstrate the accuracy and efficiency of the new compact difference scheme, compared to the standard second‐order Crank‐Nicolson scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 846–857, 2003. 相似文献
6.
Jincun Liu 《Numerical Functional Analysis & Optimization》2013,34(15):1635-1655
AbstractIn this paper, a Crank–Nicolson finite difference/finite element method is considered to obtain the numerical solution for a time fractional Sobolev equation. Firstly, the classical finite element method is presented. Stability and error estimation for the fully discrete scheme are rigorously established. However, the amount of calculation and computing time are too large due to many degrees of freedom of classical finite element scheme and nonlocality of fractional differential operator. And then the modified reduced-order finite element scheme with low dimensions and sufficiently high accuracy, which is based on proper orthogonal decomposition technique, is provided. Stability and convergence for the reduced-order scheme are also studied. At last, numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions. 相似文献
7.
A stabilized finite element method for the time-dependent Stokes equations based on Crank–Nicolson scheme is considered in this paper. The method combines the Crank–Nicolson scheme with a stabilized finite element method which uses the lowest equal-order element pair, i.e., the stabilized finite element method is applied for the spatial approximation and the time discretization is based on the Crank–Nicolson scheme. Moreover, we present optimal error estimates and prove that the scheme is unconditionally stable and convergent. Finally, numerical tests confirm the theoretical results of the presented method. 相似文献
8.
The mathematical modeling of a planar solid‐liquid interface in the solidification of a dilute binary alloy is formulating by one of nonintegrable, nonlinear evolution equation known as Sivashinsky equation. In the first part of this paper, the mathematical modeling of Sivashinsky equation is briefly discussed. Since, the exact solutions of this equation is yet unknown, obtaining its numerical solution plays an important role to simulate its behavior. Therefore, in the second part, a second‐order splitting finite difference scheme, based on Crank‐Nicolson method, is investigated to approximate the solution of the Sivashinsky equation with homogeneous boundary conditions. We prove the solvability of the present scheme and establish the error estimate of the numerical scheme. 相似文献
9.
In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
10.
Xiaoming He Tao Lin Yanping Lin Xu Zhang 《Numerical Methods for Partial Differential Equations》2013,29(2):619-646
This article presents three Crank‐Nicolson‐type immersed finite element (IFE) methods for solving parabolic equations whose diffusion coefficient is discontinuous across a time dependent interface. These methods can use a fixed mesh because IFEs can handle interface jump conditions without requiring the mesh to be aligned with the interface. These methods will be compared analytically in the sense of accuracy and computational cost. Numerical examples are provided to demonstrate features of these three IFE methods. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
11.
《Mathematical Methods in the Applied Sciences》2018,41(9):3370-3391
In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose in the first algorithm and in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly. 相似文献
12.
We prove a long-time stability result for the finite element in space, linear extrapolated Crank–Nicolson in time discretization of the Navier–Stokes equations (NSE). From this result and a numerical experiment, we show the importance of discrete mass conservation in long-time simulations of the NSE. That is, we show that using elements that strongly enforce mass conservation can provide significantly more accurate solutions over long times, compared to those that enforce it weakly. 相似文献
13.
Zhenzhen Li Dongyang Shi Minghao Li 《Mathematical Methods in the Applied Sciences》2019,42(2):605-619
In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H1‐norm and L2‐norm of the velocity and the L2‐norm of the pressure are derived. Moreover, on the basis of the optimal L2‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis. 相似文献
14.
A two‐grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
15.
《Mathematical Methods in the Applied Sciences》2018,41(7):2748-2768
Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L2‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L2 error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis. 相似文献
16.
Lalit Kumar Sivaji Ganesh Sista Konijeti Sreenadh 《Mathematical Methods in the Applied Sciences》2020,43(15):9129-9150
The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results. 相似文献
17.
《Mathematical Methods in the Applied Sciences》2018,41(2):724-749
In this article, a finite element scheme based on the Newton's method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank‐Nicolson method is used for time discretization. Well‐posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates. 相似文献
18.
Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest-order Galerkin-mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L2 -norm is in the order in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = hp = hc . This paper focuses on new and optimal error estimates of a linearized Crank–Nicolson lowest-order Galerkin-mixed finite element method (FEM), where the second-order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi-projection. Moreover, we propose a simple one-step recovery technique to obtain a new numerical Darcy velocity and pressure of second-order accuracy. Numerical results for both two and three-dimensional models are provided to confirm our theoretical analysis. 相似文献
19.
The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
20.
V. Havu 《Numerical Functional Analysis & Optimization》2013,34(7-8):825-851
We interpret the Cayley transform of linear (finite- or infinite-dimensional) state space systems as a numerical integration scheme of Crank–Nicolson type. The scheme is known as Tustin's method in the engineering literature, and it has the following important Hamiltonian integrator property: if Tustin's method is applied to a conservative (continuous time) linear system, then the resulting (discrete time) linear system is conservative in the discrete time sense. The purpose of this paper is to study the convergence of this integration scheme from the input/output point of view. 相似文献