共查询到20条相似文献,搜索用时 31 毫秒
1.
Danxia Wang Yanan Li Hongen Jia 《Numerical Methods for Partial Differential Equations》2023,39(2):1251-1265
In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H1 norm is achieved when the mesh sizes satisfy h = O(H2). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate. 相似文献
2.
A two-grid method for the elliptic equation with a small parameter ε multiplying the highest derivative is investigated. The difference schemes with the property of ε-uniform convergence on a uniform mesh and on Shishkin mesh are considered. In both cases, a two-grid method for resolving the difference scheme is investigated. A two-grid method has features that are concerned with a uniform convergence of a difference scheme. To increase the accuracy, the Richardson extrapolation in two-grid method is applied. Numerical results are discussed. 相似文献
3.
In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navier-Stokes
equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh
space and two similar linear systems (with same stiffness matrix but different right-hand side) on the fine mesh space. The
convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the
algorithm produces a numerical solution with the optimal asymptotic H
2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the
Navier-Stokes equations. 相似文献
4.
Tong Zhang 《计算数学(英文版)》2013,31(5):470-487
In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O(H2) or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0(I log hll/2H3). These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods. 相似文献
5.
A. Rathsfeld 《Mathematical Methods in the Applied Sciences》1993,16(6):443-455
In this paper we consider a quadrature method for the solution of the double-layer potential equation corresponding to Laplace's equation in a polygonal domain. We prove the stability for our method in case of special triangulations over the boundary of the polygon. For the solution of the corresponding system of linear equations, we consider a two-grid iteration and establish the rates of convergence and complexity. Finally, we discuss the effect of mesh refinement near the corners of the polygon. 相似文献
6.
7.
Guangzhi Du Qingtao Li Yuhong Zhang 《Numerical Methods for Partial Differential Equations》2020,36(6):1601-1610
In this paper, we consider the effect of adding a coarse mesh correction to the two-grid algorithm for the mixed Navier–Stokes/Darcy model. The method yields both L2 and H1 optimal velocity and piezometric head approximations and an L2 optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier–Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method. 相似文献
8.
In this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ (σ<0.5), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ/(1−σ), whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ, we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ/(1−σ). Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper. 相似文献
9.
Starting from the spectral analysis of g-circulant matrices, we study the convergence of a multigrid method for circulant and Toeplitz matrices with various size
reductions. We assume that the size n of the coefficient matrix is divisible by g≥2 such that at the lower level the system is reduced to one of size n/g, by employing g-circulant based projectors. We perform a rigorous two-grid convergence analysis in the circulant case and we extend experimentally
the results to the Toeplitz setting, by employing structure preserving projectors. The optimality of the two-grid method and
of the multigrid method is proved, when the number θ∈ℕ of recursive calls is such that 1<θ<g. The previous analysis is used also to overcome some pathological cases, in which the generating function has zeros located
at “mirror points” and the standard two-grid method with g=2 is not optimal. The numerical experiments show the correctness and applicability of the proposed ideas, both for circulant
and Toeplitz matrices. 相似文献
10.
We consider the Fourier analysis of multigrid methods (of Galerkin type) for symmetric positive definite and semi-positive
definite linear systems arising from the discretization of scalar partial differential equations (PDEs). We relate the so-called
smoothing factor to the actual two-grid convergence rate and also to the convergence rate of the V-cycle multigrid. We derive
a two-sided bound that defines an interval containing both the two-grid and V-cycle convergence rate. This interval is narrow
and away from 1 when both the smoothing factor and an additional parameter are small enough. Besides the smoothing factor,
the convergence mainly depends on the angle between the range of the prolongation and the eigenvectors of the system matrix
associated with small eigenvalues. Nice V-cycle convergence is guaranteed if the tangent of this angle has an upper bound
proportional to the eigenvalue, whereas nice two-grid convergence requires a bound proportional to the square root of the
eigenvalue. We also discuss the well-known rule which relates the order of the prolongation to that of the differential operator
associated to the problem. We first define a frequency based order which in most cases amounts to the so-called high frequency order as defined in Hemker (J Comput Appl Math 32:423–429, 1990). We give a firmer basis to the related order rule by showing that, together with the requirement of having the smoothing
factor away from one, it provides necessary and sufficient conditions for having the two-grid convergence rate away from 1. A stronger condition is further shown to be sufficient for
optimal convergence with the V-cycle. The presented results apply to rigorous Fourier analysis for regular discrete PDEs,
and also to local Fourier analysis via the discussion of semi-positive systems as may arise from the discretization of PDEs
with periodic boundary conditions. 相似文献
11.
The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh. 相似文献
12.
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
13.
In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$. 相似文献
14.
In this article, two-grid methods are studied for solving nonlinear Sobolev equation using the finite volume element method. The methods are based on one coarse grid space and one fine grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H), and the fine grid solution (with grid size h) can be obtained in a single symmetric and linear step. The optimal H1 error estimates are presented for the proposed methods, which show that the two-grid methods achieve optimal approximation as long as the mesh sizes satisfy h = 𝒪(H3|ln H|). As a result, solving such a large class of nonlinear Sobolev equations will not be much more difficult than solving one linearized equation. 相似文献
15.
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
16.
We study the solutions of block Toeplitz systems A
mn
u = b by the multigrid method (MGM). Here the block Toeplitz matrices A
mn are generated by a nonnegative function f (x,y) with zeros. Since the matrices A
mn are ill-conditioned, the convergence factor of classical iterative methods will approach 1 as the size of the matrices becomes large. These classical methods, therefore, are not applicable for solving ill-conditioned systems. The MGM is then proposed in this paper. For a class of block Toeplitz matrices, we show that the convergence factor of the two-grid method (TGM) is uniformly bounded below 1 independent of mn and the full MGM has convergence factor depending only on the number of levels. The cost per iteration for the MGM is of O(mn log mn) operations. Numerical results are given to explain the convergence rate. 相似文献
17.
Arnold Reusken 《Numerische Mathematik》1995,71(3):365-397
Summary.
We consider a two-grid method for solving 2D convection-diffusion
problems. The coarse grid correction is based on approximation of
the Schur complement. As a preconditioner of the Schur complement we use the
exact Schur complement of modified fine grid equations. We assume constant
coefficients and periodic boundary conditions and apply Fourier analysis. We
prove an upper bound for the spectral radius of the two-grid iteration
matrix that is smaller than one and independent of the mesh size, the
convection/diffusion ratio and the flow direction; i.e. we have a (strong)
robustness result. Numerical results illustrating the robustness of the
corresponding multigrid -cycle are given.
Received October 14, 1994 相似文献
18.
19.
Yidu Yang 《计算数学(英文版)》2009,27(6):748-763
This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly. 相似文献
20.
Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented. 相似文献