Two-grid Discretization Research Articles

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.

A two-level subgrid stabilized Oseen iterative method for the steady Navier–Stokes equations

Based on two-grid finite element discretization and a recent subgrid-scale model, a two-level subgrid stabilized Oseen iterative method for the convection dominated Navier–Stokes equations is proposed and analyzed. This method combines the best algorithmic features of the two-grid discretization and subgrid stabilization methods. It first solves a subgrid stabilized nonlinear Navier–Stokes problem by applying m Oseen iterations on a coarse grid, and then solves a linear problem on a finer grid where the nonlinear convection term is fixed by the coarse grid solution. Stability of the method and error estimates of the discrete solution are analyzed. The algorithmic parameter scalings are also derived. Numerical results on an example with known analytical solution, the lid-driven cavity flow and the backward-facing step flow are given to verify the theoretical predictions and demonstrate the method’s promise.

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 parallel Oseen-linearized algorithm for the stationary Navier–Stokes equations

Based on two-grid discretizations and domain decomposition, a parallel Oseen-linearized finite element algorithm for the stationary Navier–Stokes equations with moderate or large viscosity parameter is proposed and analyzed. The key idea of the algorithm is to first solve a nonlinear problem by Picard iterative method on a coarse grid, and then to solve an Oseen problem in parallel on a fine grid to correct the coarse grid solution. By using local a priori error estimate for the finite element solution and under the uniqueness condition, error bounds of the corresponding finite element solution are analyzed. Numerical results are also given to demonstrate the high efficiency of the algorithm.

A new parallel finite element algorithm for the stationary Navier–Stokes equations

Based on two-grid discretization, a new parallel finite element algorithm for the stationary Navier–Stokes equations is proposed and analyzed. This algorithm first solves the Navier–Stokes equations using a coarse grid, and then corrects the resultant residual on a fine grid by solving local Navier–Stokes equations in a parallel manner with homogeneous boundary conditions. Existing sequential Navier–Stokes solver is available for each problem on sub-domains, so that the proposed parallel algorithm can be implemented on the top of existing sequential software. The error bounds of the approximate solution are estimated. Moreover, the efficiency of the algorithm is also demonstrated by numerical simulations of the lid-driven cavity flow, the backward-facing step flow, and the flow past a circular cylinder.

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 .

Acceleration of a two-grid method for eigenvalue problems

This paper provides a new two-grid discretization method for solving partial differential equation or integral equation eigenvalue problems. In 2001, Xu and Zhou introduced a scheme that reduces the solution of an eigenvalue problem on a finite element grid to that of one single linear problem on the same grid together with a similar eigenvalue problem on a much coarser grid. By solving a slightly different linear problem on the fine grid, the new algorithm in this paper significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate. Numerical examples are also provided to demonstrate the efficiency of the new method.

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 comparison of three kinds of local and parallel finite element algorithms based on two-grid discretizations for the stationary Navier–Stokes equations

Based on two-grid discretizations, three kinds of local and parallel finite element algorithms for the stationary Navier–Stokes equations are introduced and discussed. The main technique is first to use a standard finite element discretization on a coarse grid to approximate low frequencies of the solution, then to apply some linearized discretizations on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local and parallel procedures. Three approaches to linearization are discussed. Under the uniqueness condition, error estimates of the finite element solution are derived. Numerical results show that among the three kinds of parallel algorithms, the Oseen-linearized algorithm is preferable if we both consider the computational time and the accuracy of the approximate solution.

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.

Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations

Based on two-grid discretizations, some local and parallel finite element algorithms for the d-dimensional (d = 2,3) transient Stokes equations are proposed and analyzed. Both semi- and fully discrete schemes are considered. With backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Stokes equations using a coarse grid on the entire domain, then correct the resulted residue using a finer grid on overlapped subdomains by some local and parallel procedures at each time step. By the technical tool of local a priori estimate for the fully discrete finite element solution, errors of the corresponding solutions from these algorithms are estimated. Some numerical results are also given which show that the algorithms are highly efficient.

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.

Local and parallel finite element algorithms for time-dependent convection-diffusion equations

Local and parallel finite element algorithms based on two-grid discretization for the time-dependent convection-diffusion equations are presented. These algorithms are motivated by the observation that, for a solution to the convection-diffusion problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedures. Hence, these local and parallel algorithms only involve one small original problem on the coarse mesh and some correction problems on the local fine grid. One technical tool for the analysis is the local a priori estimates that are also obtained. Some numerical examples are given to support our theoretical analysis.

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 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.

Local and parallel finite element algorithms for the stokes problem

Based on two-grid discretizations, some local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shape-regular grids.

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.

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.

A two-grid discretization method for decoupling systems of partial differential equations

In this paper, we propose a two-grid finite element method for solving coupled partial differential equations, e.g., the Schrödinger-type equation. With this method, the solution of the coupled equations on a fine grid is reduced to the solution of coupled equations on a much coarser grid together with the solution of decoupled equations on the fine grid. It is shown, both theoretically and numerically, that the resulting solution still achieves asymptotically optimal accuracy.

Local and parallel finite element algorithms based on two-grid discretization for the stream function form of Navier–Stokes equations

A new local and parallel discretization finite element algorithms are proposed and analyzed in this paper for the stream function form of Navier–Stokes equations according to [1]. The theoretical tools for analyzing these methods are some local priori estimate that are also obtained in this paper for finite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory.