Two-grid Methods for Finite Volume Element Approximations of Nonlinear Sobolev Equations

In this article, two-grid methods are studied for solving nonlinear Sobolev equation using the finite volume element method. The methods are based on one coarse grid space and one fine grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H), and the fine grid solution (with grid size h) can be obtained in a single symmetric and linear step. The optimal H¹ error estimates are presented for the proposed methods, which show that the two-grid methods achieve optimal approximation as long as the mesh sizes satisfy h = 𝒪(H³|ln H|). As a result, solving such a large class of nonlinear Sobolev equations will not be much more difficult than solving one linearized equation.     

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X_H and X_h for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ? H, respectively, and a finite element space M_h for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H⁵. If we choose H = O(h^2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.     

Two‐Grid method for nonlinear parabolic equations by expanded mixed finite element methods

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O(h^1/2). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H² for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).     

A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H¹-norm is proved to be O(h+H³|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations

A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

一类非线性对流扩散方程两重网格特征有限元方法及误差估计

主要研究了一类非线性对流扩散方程的全离散特征有限元方法的两重网格算法及其误差估计.首先在网格步长为H的粗网格上计算一个较小的非线性问题,然后利用一阶牛顿迭代和粗网格解将网格步长为h的细网格上的非线性问题转化为线性问题求解.由于非线性问题的求解仅在粗网格上进行,该两重网格算法可以节省大量的计算工作量,同时具有较高的精度,证明了该两重网格算法L~2模先验误差估计结果为O(△t+h~2+H~(4-d/2)),其中d为空间维数.     

A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.     

Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods

In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose in the first algorithm and in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly.     

Single cell finite difference approximations of O(kh2 + h4) for ∂u/∂x for one space dimensional nonlinear parabolic equation

We report a new two‐level explicit finite difference method of O(kh² + h⁴) using three spatial grid points for the numerical solution of for the solution of one‐space dimensional nonlinear parabolic partial differential equation subject to appropriate initial and Dirichlet boundary conditions. The method is shown to be unconditionally stable when applied to a linear equation. The proposed method is applicable to the problems both in cartesian and polar coordinates. Numerical examples are provided to demonstrate the efficiency and accuracy of the method discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 408–415, 2000     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Convergence analysis of a linearized Crank–Nicolson scheme for the two‐dimensional complex Ginzburg–Landau equation

A linearized Crank–Nicolson‐type scheme is proposed for the two‐dimensional complex Ginzburg–Landau equation. The scheme is proved to be unconditionally convergent in the L² ‐norm by the discrete energy method. The convergence order is \begin{align*}\mathcal{O}(\tau^2+h_1^2+h^2_2)\end{align*}, where τ is the temporal grid size and h₁,h₂ are spatial grid sizes in the x ‐ and y ‐directions, respectively. A numerical example is presented to support the theoretical result. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Two‐grid method for two‐dimensional nonlinear Schrödinger equation by finite element method

A conservative two‐grid finite element scheme is presented for the two‐dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse‐grid solution as the initial guess. Moreover, error estimates are conducted for the two‐grid method. It is shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and still achieve the asymptotically optimal approximation as long as the mesh sizes satisfy in the two‐grid method. The numerical results show that this method is very effective.