共查询到20条相似文献,搜索用时 15 毫秒
1.
The Crank–Nicolson–Galerkin finite element method for a nonlocal parabolic equation with moving boundaries
下载免费PDF全文
![点击此处可从《Numerical Methods for Partial Differential Equations》网站下载免费的PDF全文](/ch/ext_images/free.gif)
Rui M. P. Almeida José C. M. Duque Jorge Ferreira Rui J. Robalo 《Numerical Methods for Partial Differential Equations》2015,31(5):1515-1533
The aim of this article is to establish the convergence and error bounds for the fully discrete solutions of a class of nonlinear equations of reaction–diffusion nonlocal type with moving boundaries, using a linearized Crank–Nicolson–Galerkin finite element method with polynomial approximations of any degree. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with some existing moving finite element methods are investigated. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1515–1533, 2015 相似文献
2.
Dileep Kumar Sudhakar Chaudhary V.V.K. Srinivas Kumar 《Numerical Methods for Partial Differential Equations》2019,35(6):2056-2075
This article presents a finite element scheme with Newton's method for solving the time‐fractional nonlinear diffusion equation. For time discretization, we use the fractional Crank–Nicolson scheme based on backward Euler convolution quadrature. We discuss the existence‐uniqueness results for the fully discrete problem. A new discrete fractional Gronwall type inequality for the backward Euler convolution quadrature is established. A priori error estimate for the fully discrete problem in L2(Ω) norm is derived. Numerical results based on finite element scheme are provided to validate theoretical estimates on time‐fractional nonlinear Fisher equation and Huxley equation. 相似文献
3.
B. Bialecki M. Ganesh K. Mustapha 《Numerical Methods for Partial Differential Equations》2005,21(5):918-937
We propose and analyze an application of a fully discrete C2 spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H1 norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L2 and H1 norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005. 相似文献
4.
We propose, analyze, and implement fully discrete two‐time level Crank‐Nicolson methods with quadrature for solving second‐order hyperbolic initial boundary value problems. Our algorithms include a practical version of the ADI scheme of Fernandes and Fairweather [SIAM J Numer Anal 28 (1991), 1265–1281] and also generalize the methods and analyzes of Baker [SIAM J Numer Anal 13 (1976), 564–576] and Baker and Dougalis [SIAM J Numer Anal 13 (1976), 577–598]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 相似文献
5.
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
6.
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
7.
Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions 总被引:1,自引:0,他引:1
We formulate and analyze a Crank-Nicolson finite element Galerkin method and an algebraically-linear extrapolated Crank-Nicolson method for the numerical solution of a semilinear parabolic problem with nonlocal boundary conditions. For each method, optimal error estimates are derived in the maximum norm.Dedicated to Professor J. Crank on the occasion of his 80th birthdaySupported in part by the National Science Foundation grant CCR-9403461.Supported in part by project DGICYT PB95-0711. 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.
Sudhakar Chaudhary Vimal Srivastava V. V. K. Srinivas Kumar Balaji Srinivasan 《Numerical Methods for Partial Differential Equations》2017,33(3):786-813
In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657–668) for elliptic nonlocal problem of Kirchhoff type. We discuss the well‐posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for semidiscrete and fully discrete formulations in L2 and H1 norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017 相似文献
10.
Yali Gao 《Applicable analysis》2018,97(13):2288-2312
In this paper, Galerkin finite methods for two-dimensional regularized long wave and symmetric regularized long wave equation are studied. The discretization in space is by Galerkin finite element method and in time is based on linearized backward Euler formula and extrapolated Crank–Nicolson scheme. Existence and uniqueness of the numerical solutions have been shown by Brouwer fixed point theorem. The error estimates of linearlized Crank–Nicolson method for RLW and SRLW equations are also presented. Numerical experiments, including the error norms and conservation variables, verify the efficiency and accuracy of the proposed numerical schemes. 相似文献
11.
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
12.
Norbert Heuer Ernst P. Stephan 《Numerical Methods for Partial Differential Equations》1998,14(1):47-61
We propose and analyze efficient preconditioners for solving systems of equations arising from the p-version for the finite element/boundary element coupling. The first preconditioner amounts to a block Jacobi method, whereas the second one is partly given by diagonal scaling. We use the generalized minimum residual method for the solution of the linear system. For our first preconditioner, the number of iterations of the GMRES necessary to obtain a given accuracy grows like log2 p, where p is the polynomial degree of the ansatz functions. The second preconditioner, which is more easily implemented, leads to a number of iterations that behave like p log3 p. Computational results are presented to support this theory. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 47–61, 1998 相似文献
13.
Rakhim Aitbayev 《Numerical Methods for Partial Differential Equations》2008,24(2):518-534
A quadrature Galerkin scheme with the Bogner–Fox–Schmit element for a biharmonic problem on a rectangular polygon is analyzed for existence, uniqueness, and convergence of the discrete solution. It is known that a product Gaussian quadrature with at least three‐points is required to guarantee optimal order convergence in Sobolev norms. In this article, optimal order error estimates are proved for a scheme based on the product two‐point Gaussian quadrature by establishing a relation with an underdetermined orthogonal spline collocation scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
14.
Zhihao Ge Yanan He Tingting Li 《Numerical Methods for Partial Differential Equations》2019,35(4):1412-1428
In the paper, we propose a stabilized multiphysics finite element method with Crank–Nicolson scheme for a poroelasticity model. The method can eliminate the locking phenomenon and reveal the multi‐physical process. The lowest equal order finite element pair is used to reduce the computational cost. Furthermore, the method needs no constraint condition Δt = O(h2) and achieves optimal convergent order. Numerical tests are provided to illustrate the optimal accuracy and good performance in eliminating locking phenomenon of the method. 相似文献
15.
《Numerical Methods for Partial Differential Equations》2018,34(4):1282-1300
Based on overlapping domain decomposition, a new class of parallel split least‐squares (PSLS) mixed finite element methods is presented for solving parabolic problem. The algorithm is fully parallel. In the overlapping domains, the partition of unity is applied to distribute the corrections reasonably, which makes that the new method only needs one or two iteration steps to reach given accuracy at each time step while the classical Schwarz alternating methods need many iteration steps. The dependence of the convergence rate on the spacial mesh size, time increment, iteration times, and subdomains overlapping degree is analyzed. Some numerical results are reported to confirm the theoretical analysis. 相似文献
16.
Second order fully discrete defect‐correction scheme for nonstationary conduction‐convection problem at high Reynolds number
下载免费PDF全文
![点击此处可从《Numerical Methods for Partial Differential Equations》网站下载免费的PDF全文](/ch/ext_images/free.gif)
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
17.
In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H1‐norm and for pressure in L2‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes. 相似文献
18.
Chaoxia Yang 《Numerical Methods for Partial Differential Equations》2014,30(4):1279-1290
We propose a decoupled and linearized fully discrete finite element method (FEM) for the time‐dependent Ginzburg–Landau equations under the temporal gauge, where a Crank–Nicolson scheme is used for the time discretization. By carefully designing the time‐discretization scheme, we manage to prove the convergence rate , where τ is the time‐step size and r is the degree of the finite element space. Due to the degeneracy of the problem, the convergence rate in the spatial direction is one order lower than the optimal convergence rate of FEMs for parabolic equations. Numerical tests are provided to support our error analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1279–1290, 2014 相似文献
19.
Ailing Zhu Tingting Xu Qiang Xu 《Numerical Methods for Partial Differential Equations》2016,32(5):1357-1377
The semidiscrete and fully discrete weak Galerkin finite element schemes for the linear parabolic integro‐differential equations are proposed. Optimal order error estimates are established for the corresponding numerical approximations in both and norms. Numerical experiments illustrating the error behaviors are provided.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1357–1377, 2016 相似文献
20.
《Mathematical Methods in the Applied Sciences》2018,41(4):1480-1491
This paper is devoted to the analysis of a linearized theta‐Galerkin finite element method for the time‐dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on θ‐time stepping scheme with including the standard Crank‐Nicolson ( ) and the shifted Crank‐Nicolson ( , where δ is the time‐step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in L2‐norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank‐Nicolson, the shifted Crank‐Nicolson, and the general case where with k=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited. 相似文献