Two-grid Discretization Research Articles

Local and parallel finite element methods based on two-grid discretizations for the unsteady mixed Stokes-Darcy model with the Beavers-Joseph interface condition

Local and parallel finite element methods based on two-grid discretizations for the unsteady mixed Stokes-Darcy model with the Beavers-Joseph interface condition

A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods

A two-grid discretization method for nonlinear Schrödinger equation by mixed finite element methods

Local and parallel stabilized finite element methods based on two-grid discretizations for the Stokes equations

Local and parallel stabilized finite element methods based on two-grid discretizations for the Stokes equations

Local and Parallel Finite Element Methods Based on Two-grid Discretizations for a Transient Coupled Navier-Stokes/Darcy Model

Local and Parallel Finite Element Methods Based on Two-grid Discretizations for a Transient Coupled Navier-Stokes/Darcy Model

A two-grid binary level set method for structural topology optimization

A binary level set method of two-grid type is proposed for structural topology optimization. The method allows both shape and topological changes during evolution with no need for reinitialization. The Galerkin finite element method is used to discretize the linear elasticity equation, the eigenvalue problem in structure and the semi-implicit time discrete level set equation. In order to improve the efficiency of the gradient-type algorithm as well as balance the computational efforts of solvers, a nested two-grid discretization strategy is proposed. The linear elasticity or eigenvalue problem is solved on a coarse mesh, while the level set equation and the smoothing step are both discretized on a fine mesh. A variety of numerical results are given to demonstrate both the efficiency and effectiveness of the algorithm.

Local and parallel finite element methods based on two-grid discretizations for a non-stationary coupled Stokes-Darcy model

In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for a non-stationary coupled Stokes-Darcy model. Based on two-grid discretizations, a semi-discrete scheme is presented. With backward Euler scheme for the temporal discretization and two-grid discretizations for the spatial discretization, some fully discrete schemes are proposed. The crucial idea is to adopt a decoupling scheme based on interface approximation via temporal extrapolation to approximate the mixed model by utilizing a coarse grid on the whole domain, then solve some residual equations with a finer grid on some overlapped subdomains by some local and parallel procedures at each time step. The interface coupling term on the subdomains with fine grid is approximated by the coarse-grid approximations. To reach a global continuous approximation, a new parallel algorithm based on the partition of unity is devised. Some local a priori estimate that is crucial for the theoretical analysis is obtained. Finally, some numerical experiments are conducted to support our theoretical results and demonstrate the computational effectiveness.

A new symmetric mixed element method for semi-linear parabolic problem based on two-grid discretization

Based on two-grid discretization, a new numerical method is constructed for semilinear parabolic problem. In this method, a new symmetric positive definite numerical scheme is provided, and then the optimal a priori error estimates in L2 and Lp-norm are given. In order to improve the efficiency of computation, the two-grid method is considered. The corresponding convergence analysis is studied, and the optimal error estimate is derived under the relation h=O(H2). Some numerical experiments are presented to confirm our theoretical analysis.

Two-grid finite volume element method for the time-dependent Schrödinger equation

In this paper, we construct a backward Euler full-discrete two-grid finite volume element scheme for solving time-dependent Schrödinger equation. Combining the idea of the two-grid discretization, the original coupling system is solved in the coarse grid space, and the decoupling system with two independent Poisson problems is solved in the fine grid space, which ensures the accuracy and improves the computational efficiency. We further prove the optimal error estimate of the scheme rigorously. Finally, the numerical simulation results show that the two-grid method is more effective than the standard finite volume element method in solving coupled partial differential problems.

Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping

<p style='text-indent:20px;'>Using two-grid discretizations strategy, we present some local and parallel finite element algorithms for simulating the steady incompressible Navier-Stokes equations with a nonlinear damping term. In these algorithms, we compute a solution of the Navier-Stokes system with a nonlinear damping term on a coarse grid, and then adjust the solution by some local and parallel procedures on overlapped fine grid subomains. With the use of theoretical tool of local a priori estimate of the finite element solution, we estimate the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we give some numerical results to verify the theoretical predictions and demonstrate the efficiency of the proposed algorithms.</p>

Local and parallel finite element algorithms based on domain decomposition for the 2D/3D Stokes equations with damping

Based on two-grid discretization and domain decomposition approach, this paper presents and studies two local and parallel finite element algorithms for the 2D/3D steady Stokes equations with nonlinear damping term. Starting point of the algorithms is that for a solution of the Stokes equations with damping, we first approximate the low-frequency component on a relatively coarse grid, and then compute the high-frequency component on a locally fine grid via some local and parallel procedures. The proposed algorithms are easy to implement and have low communication complexity. With the use of the technical tool of local a priori estimate for finite element solution, we derive error estimates of the approximate solutions from the proposed algorithms. Some numerical results are provided to test the validity of the present algorithms.

A new two-grid mixed finite element analysis of semi-linear reaction–diffusion equation

A new mixed finite element method is established for solving the semi-linear reaction–diffusion equation based on two-grid discretization. Here, a symmetric positive definite mixed finite element method is constructed for semi-linear problem, and the corresponding error estimates in L2 and Lp-norms are considered. Then, the two-grid techniques with and without one Newton iteration correction are applied for the proposed mixed element procedure to improve the efficiency of computation. The corresponding convergence of the proposed algorithms is analyzed, and the asymptotically optimal approximation is achieved with the relation H=O(h12). Finally, numerical examples are provided to confirm theoretical results.

A novel local and parallel finite element method for the mixed Navier–Stokes–Darcy problem

In this paper, based on two-grid discretization, a new local and parallel finite element method is proposed and investigated for the mixed Navier–Stokes–Darcy problem. Compared with the classical local and parallel finite element method, the main features of our method are: (1) partition of unity method is considered to generate local subproblems; (2) global continuous approximations are achieved. We theoretically analyze the proposed method and report on some numerical tests to support the theoretical results.

Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier-Stokes equations

In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for the unsteady Navier-Stokes equations. The backward Euler scheme is considered for the temporal discretization, and two-grid method is used for the space discretization. The key idea is that for a solution to the unsteady Navier-Stokes problem, we could use a relatively coarse mesh to approximate low-frequency components and use some local fine mesh to compute high-frequency components. Some local a priori estimate is obtained. With that, theoretical results are derived. Finally, some numerical results are reported to support the theoretical findings.

Local and parallel finite element algorithms for the time-dependent Oseen equations

Based on two-grid discretizations, local and parallel finite element algorithms are proposed and analyzed for the time-dependent Oseen equations. Using conforming finite element pairs for the spatial discretization and backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Oseen equations using a coarse grid on the entire domain, and then correct the resulted residual using a fine grid on overlapped subdomains by some local and parallel procedures at each time step. By the theoretical tool of local a priori estimate for the fully discrete finite element solution, error bounds of the approximate solutions from the algorithms are estimated. Numerical results are also given to demonstrate the efficiency of the algorithms.

A Parallel Iterative Finite Element Method for the Linear Elliptic Equations

By combining the two-grid discretization with the partition of unity method, a parallel iterative finite element method for the linear elliptic equations is proposed and investigated. Since the construction of the partition of unity is based on the coarse mesh triangulation, the computational domain of each subproblem can be divided automatically and the number of subproblems can be arbitrarily huge as the coarse mesh parameter H tends to zero. That means our method can be easily implemented in high performance supercomputers or cluster of workstations. Theoretical results based on a priori error estimation of the scheme are obtained, which indicate that our method can reach the optimal convergence orders within a few two-grid iterations. Numerical results are reported to assess the theoretical results.

Local and parallel finite element methods for the coupled Stokes/Darcy model

In this paper, based on two-grid discretizations, two kinds of local and parallel finite element methods are proposed and investigated for the coupled Stokes/Darcy model. Following the idea presented in Xu and Zhou (Math. Comput. 69, 881–909 1999). a classical local and parallel finite element method is proposed and investigated. To derive global continuous approximations, a new local and parallel finite element method is devised by combining the partition of unity. We theoretically analyze the resulting formulations and derive optimal error estimates. Numerical experiments are reported to assess the theoretical results.

Expandable Parallel Finite Element Methods for Linear Elliptic Problems

In this article, two kinds of expandable parallel finite element methods, based on two-grid discretizations, are given to solve the linear elliptic problems. Compared with the classical local and parallel finite element methods, there are two attractive features of the methods shown in this article: 1) a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous; 2) the computational domain of each local subproblem is contained in a ball with radius of O(H) (H is the coarse mesh parameter), which means methods in this article are more suitable for parallel computing in a large parallel computer system. Some a priori error estimation are obtained and optimal error bounds in both H1-normal and L2-normal are derived. Finally, numerical results are reported to test and verify the feasibility and validity of our methods.

$H^2$-Conforming Methods and Two-Grid Discretizations for the Elastic Transmission Eigenvalue Problem

$H^2$-Conforming Methods and Two-Grid Discretizations for the Elastic Transmission Eigenvalue Problem

The two-grid and multigrid discretizations of the <inline-formula><tex-math id="M1">\begin{document}$ C^0 $\end{document}</tex-math></inline-formula>IPG method for biharmonic eigenvalue problem

In this paper, for the biharmonic eigenvalue problem with clamped boundary condition in \begin{document}$ \mathbb{R}^{2} $\end{document} , we study the two-grid discretization based on shifted-inverse iteration of \begin{document}$ C^0 $\end{document} IPG method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh \begin{document}$ \pi_h $\end{document} can be reduced to the solution of the eigenvalue problem on a coarser mesh \begin{document}$ \pi_H $\end{document} and the solution of a linear algebraic system on the fine mesh \begin{document}$ \pi_h $\end{document} . We prove that the resulting solution still maintains an asymptotically optimal accuracy when \begin{document}$ h\geq O(H^3) $\end{document} . In addition, we also discuss the multigrid discretization and the adaptive \begin{document}$ C^0 $\end{document} IPG algorithm based on Rayleigh quotient iteration. Numerical experiments are provided to validate the theoretical analysis.

Local and Parallel Finite Element Algorithms for the Stokes Equations with Nonlinear Slip Boundary Conditions

Based on two-grid discretizations, local and parallel finite element algorithms are studied for the Stokes equations with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind. Thereafter, the variational inequality can be transform into the variational identity as a multiplier in a convex set. The main idea of our algorithms is to approximate the low frequencies of the finite element solution using a coarse grid and use a fine grid to correct the resulted residual (that includes mostly high frequencies of the solution) by some local and parallel procedures. Error bounds for the approximate solutions are estimated. Numerical results are also given to demonstrate the effectiveness of the algorithms.