有限元二网格离散方案EQ_1^(rot)元特征值下逼近准确特征值

用数值实验结果表明把2-网格离散方案用于EQ1rot元得到的特征值给出下界。证明了当精确特征函数奇异时,对EQ1rot元使用二网格离散方案得到的特征值给出下界。

TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS

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.

Generalized Rayleigh quotient and finite element two-grid discretization schemes

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and non-conforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

TWO-GRID FINITE ELEMENT METHOD FOR TIME-FRACTIONAL NONLINEAR SCHRODINGER EQUATION

A two-grid finite element method with L1 scheme is presented for solving two-dimen-sional time-fractional nonlinear Schrodinger equation.The finite element solution in the L-norm are proved bounded without any time-step size conditions(dependent on spatial-step size).The classical L1 scheme is considered in the time direction,and the two-grid finite element method is applied in spatial direction.The optimal order error estimations of the two-grid solution in the LP-norm is proved without any time-step size conditions.It is shown,both theoretically and numerically,that the coarse space can be extremely coarse,with no loss in the order of accuracy.

A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS

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 orderα∈(1,2)andα1∈(0,1).Numerical stability and optimal error estimate O(h^(r+1)+H^(2r+2)+τ^(min{3−α,2−α1}))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.

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).

Two-Grid Crank-Nicolson FiniteVolume Element Method for the Time-Dependent Schrodinger Equation

In this paper,we construct a Crank-Nicolson finite volume element scheme and a two-grid decoupling algorithm for solving the time-dependent Schr¨odinger equation.Combining the idea of two-grid discretization,the decoupling algorithm involves solving a small coupling system on a coarse grid space and a decoupling system with two independent Poisson problems on a fine grid space,which can ensure the accuracy while the size of coarse grid is much coarser than that of fine grid.We further provide the optimal error estimate of these two schemes rigorously by using elliptic projection operator.Finally,numerical simulations are provided to verify the correctness of the theoretical analysis.

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

Two-grid mixed finite element method is proposed based on backward guler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space twogrid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.