Two-grid Method Research Articles

A unified framework for two-grid methods for a class of nonlinear problems

In this article, we present a unified error analysis of two-grid methods for a class of nonlinear problems. We first study the two-grid method of Xu by recasting the methodology in the abstract framework of Brezzi, Rappaz, and Raviart (BRR) for approximation of branches of nonsingular solutions and derive a priori error estimates. Our convergence results indicate that the correct scaling between fine and coarse meshes is given by $$h={{\mathcal {O}}}(H^2)$$ for all the nonlinear problems which can be written in and applied to the BRR framework. Next, a correction step can be added to the two-grid algorithm, which allows the choice $$h={\mathcal O}(H^3)$$ . On the other hand, the particular BRR framework with duality pairing, if it is applied to a semilinear problem, allows a higher order relation $$h={{\mathcal {O}}}(H^4)$$ . Furthermore, even the choice $$h={{\mathcal {O}}}(H^5)$$ is possible with the correction step either on fine mesh or coarse mesh. In addition, elliptic problems with gradient nonlinearities and the Naiver–Stokes equations are considered to illustrate our unified theory. Finally, numerical experiments are conducted to confirm our theoretical findings. Numerical results indicate that the correction step used as a simple postprocessing enhances the solution accuracy, particularly for problems with layers.

ABOUT WAVELET-BASED MULTIGRID NUMERICAL METHOD OF STRUCTURAL ANALYSIS WITH THE USE OF DISCRETE HAAR BASIS

he distinctive paper is devoted to so-called multigrid (particularly two-grid) method of structural analysis based on discrete Haar basis (one-dimensional, two-dimensional and three-dimensional problems are under consideration). Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Special projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the refinement component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation is a system of linear algebraic equations which is constructed with the use of finite element method or finite difference method. Block Gauss method can be used for direct solution.

Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations

In this paper, we present a second-order accurate Crank-Nicolson scheme for the two-grid finite element methods of the nonlinear Sobolev equations. This method involves solving a small nonlinear system on a coarse mesh with mesh size H and a linear system on a fine mesh with mesh size h, which can still maintain the asymptotically optimal accuracy compared with the standard finite element method. However, the two-grid scheme can reduce workload and save a lot of CPU time. The optimal error estimates in H1-norm show that the two-grid methods can achieve optimal convergence order when the mesh sizes satisfy h = O(H2). These estimates are shown to be uniform in time. Numerical results are provided to verify the theoretical estimates.

A two-grid method for incompressible miscible displacement problems by mixed finite element and Eulerian–Lagrangian localized adjoint methods

In this paper, we present a scheme for solving two-dimensional miscible displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/2) in this paper.

A two-grid decoupled algorithm for fracture models

In this paper, we consider a two-grid decoupled algorithm for two categories of fracture models. In the mixed Darcy/Darcy fracture model, fluid flow in fractures as well as in the surrounding medium is governed by Darcy’s law. And in the mixed Darcy–Forchheimer/Darcy fracture model, flow in fractures is governed by Darcy–Forchheimer’s law while that in surrounding matrix is governed by Darcy’s law. In the proposed two-grid method, we use a coarse grid approximation to the interface coupling conditions for decoupling mixed problems in fractures and surrounding matrix. Error estimates show that the two-grid decoupled algorithm retains the same order of approximation accuracy as the coupled algorithm for the pressure p on H1 semi-norm and the velocity u on (L2)2 norm. Numerical experiments are carried out to verify the accuracy and efficiency of the decoupled algorithm, in which the computation times are reduced greatly compared to the coupled method.

Two-grid methods for expanded mixed finite element approximations of semi-linear parabolic integro-differential equations

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h=H2, the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method.

Superconvergence analysis of a two-grid method for semilinear parabolic equations

In this paper, the superconvergence analysis of a two-grid method (TGM) is established for the semilinear parabolic equations. Based on the combination of the interpolation and Ritz projection technique, an important ingredient in the method, the superclose estimates in the H 1 -norm are deduced for the backward Euler fully-discrete TGM scheme. Moreover, through the interpolated postprocessing approach, the corresponding global superconvergence result is derived. Finally, some numerical results are provided to confirm the theoretical analysis, and also show that the computing cost of the proposed TGM is only half of the conventional Galerkin finite element methods (FEMs).

LOCAL NUMERICAL SOLUTION OF BOUNDARY PROBLEMS OF STRUCTURAL MECHANICS WITH THE USE OF HYBRID DISCRETE WAVELET BASIS

The distinctive paper is devoted to the two-grid method of structural analysis based on hybrid discrete wavelet basis (in particular, the simplest one-dimensional problems are under consideration). Approximations of he mesh functions in this basis, general idea of the two-grid method and one-dimensional numerical example of beam analysis are presented.

Two-grid method for compressible miscible displacement problem by CFEM–MFEM

This paper is concerned about the error analysis of two-grid method for compressible miscible displacement in porous medium. A characteristics finite element method (CFEM) is presented for the concentration equation to handle the convection part, and standard mixed finite element method (MFEM) is used for the pressure equation. Moreover, we linearize the equations based on the Newton iteration method, then, two-grid method is considered in this full discrete scheme problem. We prove the Lp error estimates for the pressure, Darcy velocity, concentration variables in the two-grid method. It is shown that coarse space can be extremely coarse and we achieve asymptotically optimal approximation. Finally, numerical experiment indicates that two-grid method is a very effective method for solving miscible displacement problem.

Optimal Interpolation and Compatible Relaxation in Classical Algebraic Multigrid

In this paper, we consider a classical algebraic multigrid (AMG) form of optimal interpolation that directly minimizes the two-grid convergence rate and compare it with a so-called ideal interpolation that minimizes a weak approximation property of the coarse space. We study compatible relaxation type estimates for the quality of the coarse grid and derive a new sharp measure using optimal interpolation that provides a guaranteed lower bound on the convergence rate of the resulting two-grid method for a given grid. In addition, we design a generalized bootstrap AMG setup algorithm that computes a sparse approximation to the optimal interpolation matrix. We demonstrate numerically that the bootstrap AMG method with sparse interpolation matrix (and spanning multiple levels) converges faster than the two-grid method with the standard ideal interpolation (a dense matrix) for various scalar diffusion problems with highly varying diffusion coefficient.

Two-Grid and Multiple-Grid Arnoldi for Eigenvalues

A new approach is given for computing eigenvalues and eigenvectors of large matrices. Multiple grids are combined with the Arnoldi method in order to solve difficult problems. First, a two-grid method computes eigenvectors on a coarse grid and improves them on the fine grid. On the fine grid, an Arnoldi-type method is used that, unlike standard Arnoldi methods, can accept initial approximate eigenvectors. This two-grid approach can vastly improve Krylov computations. It also succeeds for problems that regular multigrid cannot handle. Analysis is given to explain why fine grid convergence rates can sometimes unexpectedly match that of regular Arnoldi on the fine grid in spite of having many initial guess vectors. Near-Krylov theory is developed for this. Finally, this new method is generalized for more grid levels to produce a multiple-grid Arnoldi method.

Error Estimates of the Classical and Improved Two-Grid Methods

Error Estimates of the Classical and Improved Two-Grid Methods

WO-GRID METHOD OF STRUCTURAL ANALYSIS BASED ON DISCRETE HAAR BASIS. PART 1: ONE-DIMENSIONAL PROBLEMS

The distinctive paper is devoted to the two-grid method of structural analysis based on discrete Haar basis (in particular, the simplest one-dimensional problems are under consideration). A brief review of publications of recent years of Russian and foreign specialists devoted to the current trends in the use of wavelet analysis in construction mechanics is given. Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the detailing component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation (defined on a given interval) is a system of linear algebraic equations (SLAE) constructed within finite difference method (FDM) or the finite element method (FEM). Next, the transition to the resolving SLAE is done. Block Gauss method is used for its direct solution (forward-backward algorithm is realized). We consider a numerical solution of the boundary problem of bending of the Bernoulli beam lying on an elastic foundation (within Winkler model) as a practically important one-dimensional sample. There is good consistency of the results obtained by the proposed method and by standard finite difference method.

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

A conservative two-grid mixed finite element scheme is presented for 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 H=O(h12) in the two-grid method. The numerical results show that this method is very effective.

Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank–Nicolson scheme and the so-called weighted and shifted Grünwald formula.By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate.

Two-grid mixed finite element method for nonlinear hyperbolic equations

In this paper, a combination method of mixed finite element method and two-grid scheme is constructed for solving numerically the two-dimensional nonlinear hyperbolic equations. 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. 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 H=O(h12) in the first algorithm and H=O(h14) in the second algorithm, respectively. Error estimates are derived in detail. Finally, two numerical examples are carried out to verify the accuracy and efficiency of the presented method.

A Two-Level Sparse Grid Collocation Method for Semilinear Stochastic Elliptic Equation

Abstract In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ⁢ ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ⁢ ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} (where η ⁢ ( L , N ) , η ⁢ ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ⁢ ( ℓ , N ) ≈ ( η ⁢ ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ⁢ ( Γ ) ⊗ ℒ 2 ⁢ ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ⁢ ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} in random spaces.

Stability analysis of nonlinear two-grid method for multigroup neutron diffusion problems

We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.

Two-grid methods for a class of nonlinear elliptic eigenvalue problems

In this article, we introduce and analyse some two-grid methods for nonlinear elliptic eigenvalue problems of the form div(Du)+Vu+f(u2)u=»u,%u%L2=1. We provide a priori error estimates for the ground state energy, the eigenvalue » and the eigenfunction u, in various Sobolev norms. We focus in particular on the Fourier spectral approximation (for periodic boundary conditions), and on the P1 and P2 finite element discretizations (for homogeneous Dirichlet boundary conditions), taking numerical integration errors into account. Finally, we provide numerical examples illustrating our analysis.

DEM–CFD simulation of modular PB-FHR core with two-grid method

For designing and optimizing the reactor core of modular pebble-bed fluoride salt-cooled high-temperature reactor (PB-FHR), it is of importance to simulate the coupled fluid and particle flow due to strong coolant–pebble interactions. Computational fluid dynamics and discrete element method (DEM) coupling approach can be used to track particles individually while it requires a fluid cell being greater than the pebble diameter. However, the large size of pebbles makes the fluid grid too coarse to capture the complicated flow pattern. To solve this problem, a two-grid approach is proposed to calculate interphase momentum transfer between pebbles and coolant without the constraint on the shape and size of fluid meshes. The solid velocity, fluid velocity, fluid pressure and void fraction are mapped between hexahedral coarse particle grid and fine fluid grid. Then the total interphase force can be calculated independently to speed up computation. To evaluate suitability of this two-grid approach, the pressure drop and minimum fluidization velocity of a fluidized bed were predicted, and movements of the pebbles in complex flow field were studied experimentally and numerically. The spouting fluid through a central inlet pipe of a scaled visible PB-FHR core facility was set up to provide the complex flow field. Water was chosen as liquid to simulate the molten salt coolant, and polypropylene balls were used to simulate the pebble fuels. Results show that the pebble flow pattern captured from experiment agrees well with the simulation from two-grid approach, hence the applicability of the two-grid approach for the later PB-FHR core design.