Two-grid Discretization Scheme Research Articles

Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem

In this paper, for a new Stekloff eigenvalue problem which is non-selfadjoint and not H1-elliptic, we establish and analyze two kinds of two-grid discretization scheme and a local finite element scheme. We present the error estimates of approximations of two-grid discretizations. We also prove a local error estimate which is suitable for the case that the local refined region contains singular points lying on the boundary of domain. Numerical experiments are reported finally to show the efficiency of our schemes.

A two-grid discretization scheme of non-conforming finite elements for transmission eigenvalues

In this paper, for the Helmholtz transmission eigenvalue problem, we propose a two-grid discretization scheme of non-conforming finite elements. With this scheme, the solution of the transmission eigenvalue problem on a fine grid πh is reduced to the solution of the primal and dual eigenvalue problem on a much coarser grid πH and the solutions of two linear algebraic systems with the same positive definite Hermitian and block diagonal coefficient matrix on the fine grid πh. We prove the resulting solution still maintains an asymptotically optimal accuracy, and we report some numerical examples in two dimension and three dimension on the modified-Zienkiewicz element to validate the efficiency of our approach for solving transmission eigenvalues.

A two-grid discretization scheme for optimal control problems of elliptic equations

In the paper, a two-grid finite element scheme is discussed for distributed optimal control governed by elliptic equations. With this new scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.

Two-Grid Discretization Scheme for Nonlinear Reaction Diffusion Equation by Mixed Finite Element Methods

AbstractIn this paper, we study an efficient scheme for nonlinear reaction-diffusion equations discretized by mixed finite element methods. We mainly concern the case when pressure coefficients and source terms are nonlinear. To linearize the nonlinear mixed equations, we use the two-grid algorithm. We first solve the nonlinear equations on the coarse grid, then, on the fine mesh, we solve a linearized problem using Newton iteration once. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfyH=O(h1/2). As a result, solving such a large class of nonlinear equations will not be much more difficult than getting solutions of one linearized system.

A Two-Grid Method of Nonconforming Element Based on the Shifted-Inverse Power Method for the Steklov Eigenvalue Problem

This paper establishes a new kind of two-grid discretization scheme of nonconforming Crouzeix-Raviart element based on the shifted-inverse power method for the Steklov eigenvalue problem. The error estimates are provided from the work of Yang and Bi (SIAM J. Numer. Anal., 49, pp.1602-1624, 2011). Finally, numerical experiments are reported to illustrate the high efficiency of the two-grid discretization scheme proposed in this paper.

The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

In this paper, adaptive finite element methods for differential operator eigenvalue problems are discussed. For multi-scale discretization schemes based on Rayleigh quotient iteration (see Scheme 3 in [Y. Yang, H. Bi, A two-grid discretization scheme based on shifted-inverse power method, SIAM J. Numer. Anal. 49 (2011) 1602–1624]), a reliable and efficient a posteriori error indicator is given, in addition, a new adaptive algorithm based on the multi-scale discretizations is proposed, and we apply the algorithm to the Schrödinger equation for hydrogen atoms. The algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

The Efficient Discretization Schemes for the Maxwell Eigenvalue Problem

In this paper, we explore efficient discretization schemes—two kinds of two-grid discretization schemes based on the parameterized approach for the Maxwell eigenvalue problem. We give numerical experiments by using Matlab to write program, and show that this algorithm is highly effective.

A two-grid method of the non-conforming Crouzeix–Raviart element for the Steklov eigenvalue problem

This paper discusses a high efficient scheme for the Steklov eigenvalue problem. A two-grid discretization scheme of nonconforming Crouzeix–Raviart element is established. With this scheme, the solution of a Steklov eigenvalue problem on a fine grid π h is reduced to the solution of the eigenvalue problem on a much coarser grid π H and the solution of a linear algebraic system on the fine grid π h . By using spectral approximation theory and Nitsche–Lascaux–Lesaint technique in space H - 1 2 ( ∂ Ω ) , we prove that the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking H = h . And the numerical experiments indicate that when the eigenvalues λ k, h of nonconforming Crouzeix–Raviart element approximate the exact eigenvalues from below, the approximate eigenvalues λ k , h ∗ obtained by the two-grid discretization scheme also approximate the exact ones from below, and the accuracy of λ k , h ∗ is higher than that of λ k, h .

Two-Grid Finite Element Discretization Schemes Based on Shifted-Inverse Power Method for Elliptic Eigenvalue Problems

This paper discusses highly efficient discretization schemes for solving self-adjoint elliptic differential operator eigenvalue problems. Several new two-grid discretization schemes, including the conforming and nonconforming finite element schemes, are proposed by combining the finite element method with the shifted-inverse power method for matrix eigenvalue problems. With these schemes, the solution of an eigenvalue problem on a fine grid $\pi_{h}$ is reduced to the solution of an eigenvalue problem on a much coarser grid $\pi_{H}$ and the solution of a linear algebraic system on the fine grid $\pi_{h}$. Theoretical analysis shows that the schemes have a high efficiency. For instance, the resulting solution can maintain an asymptotically optimal accuracy by using the conforming linear element or the nonconforming Crouzeix–Raviart element by taking $H=O(\sqrt[4]{h})$. Numerical experiments are presented to support the theoretical analysis. In addition, this paper establishes multigrid discretization schemes and proves their efficiency.

A two-grid discretization scheme for the Steklov eigenvalue problem

In the paper, a two-grid discretization scheme is discussed for the Steklov eigenvalue problem. With the scheme, the solution of the Steklov eigenvalue problem on a fine grid is reduced to the solution of the Steklov eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. Using spectral approximation theory, it is shown theoretically that the two-scale scheme is efficient and the approximate solution obtained by the scheme maintains the asymptotically optimal accuracy. Finally, numerical experiments are carried out to confirm the considered theory.

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 nonconforming 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 Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems

This paper extends the two-grid discretization scheme of the conforming flnite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming flnite 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 flne mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the flne mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming flnite 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 flrst eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the flne mesh directly.

An adaptive multigrid scheme for Bose–Einstein condensates in a periodic potential

We present a novel multigrid-continuation method for treating parameter-dependent problems. The proposed algorithm which can be flexibly implemented is a generalization of the two-grid discretization schemes [C.-S. Chien, B.-W. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput. 27 (2006) 1287–1304]. That is, approximating points on a solution curve do not necessarily lie on the same fine grid. We apply the algorithm to compute energy levels and superfluid densities of Bose–Einstein condensates (BEC) in a periodic potential. Both positive and negative scattering lengths are considered in our numerical experiments. For positive scattering length, if the chemical potential is large enough, and the domain is properly chosen, the results show that the number of peaks of the first few energy states of the 2D BEC in a periodic potential depends on the wave number of the periodic potential. Moreover, for bright solitons the number of peaks of the ground state solutions is ( 1 d − 1 ) 2 and ( 1 d ) 2 , where the periodic potential is expressed in terms of the sine or the cosine functions, respectively. However, these formulae do not hold if the scattering length is negative. The numerical study is extended to the two-component, 1D and 2D BEC in a periodic potential.

Two-grid discretization schemes for nonlinear Schrödinger equations

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.

A two-grid discretization scheme for eigenvalue problems

A two-grid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.