Two-grid Finite Element Method Research Articles

Fast Two-Grid Finite Element Algorithm for a Fractional Klein- Gordon Equation

In this article, we propose a spatial two-grid finite element algorithm combined with a shifted convolution quadrature (SCQ) formula for solving the fractional Klein-Gordon equation. The time direction at tn − θ is approximated utilizing a second-order SCQ formula, where θ is an arbitrary constant. The spatial discretization is performed using a two-grid finite element method involving three steps: calculating the numerical solution by solving a nonlinear system iteratively on the coarse grid, obtaining the interpolation solution based on the computed solutions in the first step, and solving a linear finite element system on the fine grid. We present a numerical algorithm, validate the two-grid finite element method’s effectiveness, and demonstrate the computational efficiency for our method by the comparison of the computing results between the two-grid finite element method and the standard finite element method.

Superconvergence analysis of a two-grid BDF2-FEM for nonlinear dispersive wave equation

The aim of this paper is to study the superconvergent behavior of a two-grid finite element method (FEM) with 2-step backward differential formula (BDF2) for nonlinear dispersive wave equation. By introducing an auxiliary variable q=ut for the original variable u, the problem is transformed into a parabolic system. With the help of the combination technique of the interpolation and Ritz projection, the superclose and superconvergent estimates for the above two variables in H1-norm are obtained under the lower regularity of the exact solution compared with the standard Galerkin FEM. Finally, some numerical results are provided to show the correctness of the theoretical predictions and good performance of the proposed method.

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>

Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

Numerical analysis of two-dimensional unsaturated soil water flow problems with two-grid finite element methods

<p style='text-indent:20px;'>In this paper, we investigate a two-grid finite element method (FEM) for two-dimensional nonlinear unsaturated soil water flow problems. The application of two-grid methods can transform the solution of nonlinear problems on the fine grid to the solution of nonlinear problems on the coarse grid and the solution of linear problems on the fine grid. Based on the error results of standard FEM, we derive the continuous and discrete time error estimates in the energy norm for the two-grid method. Numerical experiments are carried out with the linear element as an example to illustrate that the two-grid method can achieve the similar optimal accuracy as standard FEM with less time as long as the two mesh sizes satisfy <inline-formula><tex-math id="M1">\begin{document}$ H = O(\sqrt{h}) $\end{document}</tex-math></inline-formula>.</p>

Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations

Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations

Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations

In this article, an efficient two-grid finite element method is proposed for solving the nonlinear time fractional variable coefficient diffusion equations. This algorithm firstly solves a nonlinear system to get the numerical solution uHn on the coarse grid with size H, then based on the initial iterative solution uHn on the coarse grid, the linearized finite element system is solved on the fine grid with size h to get the numerical solution Uhn, in which the temporal direction is approximated by the L2−1σ scheme. Besides, the stability and priori error estimates of standard finite element method and two-grid method are given. Finally, the validity and efficiency of the two-grid algorithm are verified by two numerical experiments.

A fully discrete two-grid method for the diffusive Peterlin viscoelastic model

We present in this paper a fully discrete two-grid finite element method to solve the diffusive Peterlin viscoelastic model on the basis of studying the fully discrete standard finite element method. The fully discrete two-grid method involves solving small nonlinear coarse spatial grid systems and linear fine spatial grid systems with the same time step. Under the reasonable assumptions, it is proven theoretically that the accuracy of fully discrete two-grid method is the same as the fully discrete finite element method on the fine spatial grid. Numerical examples are given to validate the accuracy and convergence order of the standard finite element method and two-grid method.

A Time Two-Mesh Compact Difference Method for the One-Dimensional Nonlinear Schrödinger Equation.

The nonlinear Schrödinger equation is an important model equation in the study of quantum states of physical systems. To improve the computing efficiency, a fast algorithm based on the time two-mesh high-order compact difference scheme for solving the nonlinear Schrödinger equation is studied. The fourth-order compact difference scheme is used to approximate the spatial derivatives and the time two-mesh method is designed for efficiently solving the resulting nonlinear system. Comparing to the existing time two-mesh algorithm, the novelty of the new algorithm is that the fine mesh solution, which becomes available, is also used as the initial guess of the linear system, which can improve the calculation accuracy of fine mesh solutions. Compared to the two-grid finite element methods (or finite difference methods) for nonlinear Schrödinger equations, the numerical calculation of this method is relatively simple, and its two-mesh algorithm is implemented in the temporal direction. Taking advantage of the discrete energy, the result with in the discrete -norm is obtained. Here, and are the temporal parameters on the coarse and fine mesh, respectively, and h is the space step size. Finally, some numerical experiments are conducted to demonstrate its efficiency and accuracy. The numerical results show that the new algorithm gives highly accurate results and preserves conservation laws of charge and energy. Furthermore, by comparing with the standard nonlinear implicit compact difference scheme, it can reduce the CPU time without loss of accuracy.

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.

Two-grid finite element methods for nonlinear time-fractional parabolic equations

Two-grid finite element methods for nonlinear time-fractional parabolic equations

A fully discrete two-grid finite element method for nonlinear hyperbolic integro-differential equation

Two-grid fully discrete finite element approximations of the solution for a nonlinear hyperbolic integro-differential equation are considered and analyzed in this paper. The H1-norm error estimate is derived, which shows that the optimal convergence order can be obtained when the coarse-grid of size H and the fine-grid of size h satisfy h=O(H2). Besides reducing the storage and saving a large amount of time, two-grid methods also keep the accuracy of convergence in calculations. Numerical examples are given to support our theoretical results and demonstrate the efficiency of the methods.

Superconvergence analysis for nonlinear Schrödinger equation with two-grid finite element method

In this paper, two-grid finite element method is applied to solve the nonlinear Schrödinger equation. On one hand, prior estimates of the numerical solution on the coarse grid and the introduced temporal equation are derived. Subsequently, taking advantage of the mathematical induction method and an identical equation of the inner product, result with order O(h2+τ) in L2-norm is deduced on the coarse grid and unconditional boundedness of the numerical solution is deduced as a result. Unconditional superclose result with order O(h2+τ) in H1-norm is arrived at based on the above results. On the other hand, on the fine grid, an Euler scheme is constructed based on the numerical solution of the coarse grid. To achieve unconditional superconvergent estimate of order O(H4+h2+τ) in H1-norm, we give fine estimate of the nonlinear term with H2=O(h). At last, numerical example is introduced to confirm the theoretical analysis. Here, H and h are the spatial parameters on the coarse grid and the fine grid, respectively, and τ is the time step.

Two-grid finite element method for the dual-permeability-Stokes fluid flow model

In this paper, two-grid finite element method for the steady dual-permeability-Stokes fluid flow model is proposed and analyzed. Dual-permeability-Stokes interface system has vast applications in many areas such as hydrocarbon recovery process, especially in hydraulically fractured tight/shale oil/gas reservoirs. Two-grid method is popular and convenient to solve a large multiphysics interface system by decoupling the coupled problem into several subproblems. Herein, the two-grid approach is used to reduce the coding task substantially, which provides computational flexibility without losing the approximate accuracy. Firstly, we solve a global problem through standard Pk − Pk− 1 − Pk − Pk finite elements on the coarse grid. After that, a coarse grid solution is applied for the decoupling between the interface terms and the mass exchange terms to solve three independent subproblems on the fine grid. The three independent parallel subproblems are the Stokes equations, the microfracture equations, and the matrix equations, respectively. Four numerical tests are presented to validate the numerical methods and illustrate the features of the dual-permeability-Stokes model.

Two-Grid Finite Element Methods of Crank-Nicolson Galerkin Approximation for a Nonlinear Parabolic Equation

Two-Grid Finite Element Methods of Crank-Nicolson Galerkin Approximation for a Nonlinear Parabolic Equation

Unconditional superconvergence analysis of a two-grid finite element method for nonlinear wave equations

This paper aims to present a two-grid method (TGM) with low order nonconforming EQ1rot finite element for solving a class of nonlinear wave equations, and to give the superconvergent error analysis unconditionally. Firstly, for the Crank-Nicolson fully discrete scheme, the existence and uniqueness of the numerical solutions are proved, and based on the special characters of this element as well as the priori estimates, the supercloseness and superconvergence analysis of both the original variable u and the auxiliary variable q=ut in the broken H1-norm are deduced on the coarse mesh for the Galerkin finite element method (FEM). Then, by employing the interpolation postprocessing approach and the boundness of the numerical solution in the broken H1-norm on the coarse mesh, the corresponding superconvergence results of order O(τ2+H4+h2) for the TGM are obtained unconditionally, here τ, H and h denote time step, coarse and fine grid sizes, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.

Superconvergence analysis of two-grid finite element method for nonlinear Benjamin-Bona-Mahony equation

In this paper, a new second-order accurate backward differentiation formula of the two-grid finite element method is developed for the nonlinear Benjamin-Bona-Mahony equation, and the stability and unconditional superconvergence analysis of this scheme are presented. This method firstly solves a nonlinear system on a coarse mesh with size H to get the numerical solution UHn and then computes a linear system constructed by the previous result UHn on a fine mesh with size h(h<<H) to get the numerical solution Uhn. The new scheme has two big advantages. One is that the same superclose and superconvergence behavior of the H1-norm can be remained compared with the standard finite element methods, when the mesh sizes satisfy h=O(H2). The other is that more than two thirds of the computational cost can be saved for the test examples. Two numerical experiments are provided to verify the theoretical analysis.

Numerical solution of two-dimensional nonlinear Schrödinger equation using a new two-grid finite element method

A new two-grid finite element scheme is presented for two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the nonlinear system using the coarse-grid solution as the initial guess, and furthermore one more linear system on the coarse space is solved. The error estimations of the two-grid solution in the L2 and H1 norm are given. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and two-grid algorithm still achieves optimal approximation as long as the mesh sizes satisfy H=O(h13).

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in H1-norm and for pressure in L2-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.

A two-grid finite element method for nonlinear parabolic integro-differential equations

ABSTRACTIn this paper, we present a two-grid finite element method (FEM) for a two-dimensional nonlinear parabolic integro-differential equation. We solve a fully nonlinear system on a coarse grid space with a grid size H and derive a rough approximation of the exact solution, and then solve the corresponding linearized problem on a fine grid space with a grid size h. The optimal error estimates in -norm are obtained for spatially the semidiscrete two-grid FEM. Finally, numerical examples are given to testify the efficiency of the method.