Two-grid Finite Element Scheme Research Articles

Two-grid algorithm of lumped mass finite element approximation for Allen-Cahn equations

In this paper, we present a two-grid lumped mass finite element algorithm for 2D Allen-Cahn equations, where Crank-Nicolson scheme and piecewise linear element are utilized for temporal and spatial discretization, respectively. Both the maximum-norm boundedness and H1-norm error estimates of the proposed two-grid scheme are discussed. Finally, a numerical example is given to verify the theoretical results. By comparing with the finite element scheme and the two-grid finite element scheme, we found that our scheme has a great advantage in calculation time.

Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations

<p style='text-indent:20px;'>In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the <inline-formula><tex-math id="M1">$ H^1 $</tex-math></inline-formula>-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.</p>

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).

Two-grid method for the two-dimensional time-dependent Schrödinger equation by the finite element method

In this paper, we construct a backward Euler full-discrete two-grid finite element scheme for the two-dimensional time-dependent Schrödinger equation. With this method, the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two Poisson equations on the same fine grid. We analyze the error estimate of the standard finite element solution and the two-grid solution in the H1 norm. It is shown that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(hkk+1). Finally, a numerical experiment indicates that our two-grid algorithm is more efficient than the standard finite element method.

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.

An expandable local and parallel two-grid finite element scheme

An expandable local and parallel two-grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of Poisson equation. Compared with the usual local and parallel finite element schemes, the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of CPUs. Convergence results based on H1 and L2 a priori error estimation of the scheme are obtained, which show that the scheme can reach the optimal convergence orders within |lnH|2 or |lnH| two-grid iterations if the coarse mesh size H and the fine mesh size h are properly configured in 2-D or 3-D case, respectively. Some numerical results are presented at the end of the paper to support our analysis.

On a two-grid finite element scheme combined with Crank–Nicolson method for the equations of motion arising in the Kelvin–Voigt model

In this paper, we present a second order accurate Crank–Nicolson scheme applied to a system arising from Galerkin finite element two-grid approximations to the equations of motion described by the Kelvin–Voigt viscoelastic fluid flow model. Optimal error estimates in L∞(L2)-norm and L∞(H1)-norm for velocity and L∞(L2)-norm for pressure are established. These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.

On a two-grid finite element scheme for the equations of motion arising in Kelvin-Voigt model

In this paper, we study a two level method based on Newton's iteration for the nonlinear system arising from the Galerkin finite element approximation to the equations of motion described by the Kelvin-Voigt viscoelastic fluid flow model. The two-grid algorithm is based on three steps. In the first step, the nonlinear system is solved on a coarse mesh 𝒯 H $\mathcal {T}_{H}$ to obtain an approximate solution u H . In the second step, the nonlinear system is linearized around u H based on Newton's iteration and the linear system is solved on a finer mesh 𝒯 h $\mathcal {T}_{h}$ . Finally, in the third step, a correction to the results obtained in the second step is achieved by solving a linear problem with a different right hand side on 𝒯 h $\mathcal {T}_{h}$ . Optimal error estimates in L ? (L 2)-norm, when h = 𝒪 ( H 2 ? ? ) $h=\mathcal {O} (H^{2-\delta })$ and in L ? (1)-norm, when h = 𝒪 ( H 5 ? 2 ? ) $h=\mathcal {O}(H^{5-2\delta })$ for velocity and in L ? (L 2)-norm, when h = 𝒪 ( H 5 ? 2 ? ) $h=\mathcal {O}(H^{5-2\delta })$ for pressure are established, where ? > 0 is arbitrarily small for two dimensions and ? = 1 2 $\delta =\frac {1}{2}$ for three dimensions.

A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation

We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of sizeH. The nonlinearities are expanded about the coarse grid solution on the fine gird of sizeh. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with theH1-normO(h+H2)for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfyh=O(H2).

Two-grid finite element scheme for the fully discrete time-dependent Navier–Stokes problem

In this Note, we study a two-grid scheme fully discrete in time and space for solving the Navier–Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity u H computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of u H to the error in the non-linear term, is measured in the L 2 norm in space and time, and thus has hopefully a higher-order than if it were measured in the H 1 norm in space. We present the following results: if h = H 2 = k , then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid. To cite this article: H. Abboud et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).