Two-grid Method Research Articles

Non-Intrusive Reduced Basis two-grid method for flow and transport problems in heterogeneous porous media

Due to its non-intrusive nature and ease of implementation, the Non-Intrusive Reduced Basis (NIRB) two-grid method has gained significant popularity in numerical computational fluid dynamics simulations. The efficiency of the NIRB method hinges on separating the procedure into offline and online stages. In the offline stage, a set of high-fidelity computations is performed to construct the reduced basis functions, which is time-consuming but is only executed once. In contrast, the online stage adapts a coarse-grid model to retrieve the expansion coefficients of the reduced basis functions. Thus it is much less costly than directly solving a high-fidelity model. However, coarse grids in heterogeneous porous media of flow models are often accompanied by upscaled hydraulic parameters (e.g. hydraulic conductivity), thus introducing upscaling errors. In this work, we introduce the two-scale idea to the existing NIRB two-grid method: when dealing with coarse-grid models, we also employ upscaled model parameters. Both the discretization and upscaling errors are compensated by the rectification post-processing. The numerical examples involve flow and heat transport problems in heterogeneous hydraulic conductivity fields, which are generated by self-affine random fields. Our research findings indicate that the modified NIRB method can effectively capture the large-scale features of numerical solutions, including pressure, velocity, and temperature. However, accurately retrieving velocity fields with small-scale features remains highly challenging.

Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods

Abstract In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H H and the fine grid size h h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) ( k ≥ 1 k\ge 1 ), where k k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

A reduced-order two-grid method based on POD technique for the semilinear parabolic equation

In the conventional two-grid (TG) method, the nonlinear system on the fine grid is transformed into a nonlinear subsystem on the coarse grid and a linear subsystem on the fine grid to reduce computational costs. It has been successfully applied in various fields. Nonetheless, its computational efficiency remains relatively low. For this, we develop a novel reduced-order two-grid (ROTG) method with less degrees of freedom for solving the semilinear parabolic equation. For the two subsystems mentioned, the proper orthogonal decomposition (POD) technique is utilized to substantially reduce degrees of freedom. An a priori error estimate for the ROTG scheme is derived. Finally, we conduct several numerical tests to observe the ROTG method's behavior and verify the theoretical analysis.

Construction of a new time-space two-grid method and its solution for the generalized Burgers’ equation

In this paper, we construct a new time-space two-grid method and apply it to the generalized Burgers’ (GB) equation. We prove that our proposed scheme is solvable, energy-preserving and convergent in the discrete L2-norm. Compared with previous studies on the two-grid method, the new time-space two-grid technique allows for a more flexible selection of the coarse grid in both temporal and spatial dimensions, resulting in greater computational efficiency and a simpler mesh selection. Finally, we compare our scheme with the general difference method (Zhang et al., 2021) and the time two-grid method (Shi and Yang, 2024), revealing that the proposed new time-space two-grid method is more efficient in computational time.

A Note on the Convergence of Multigrid Methods for the Riesz–Space Equation and an Application to Image Deblurring

In recent decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz–Space FDE whose theoretical convergence analysis of such multigrid methods is currently limited in the relevant literature to the two-grid method. Here we provide a detailed theoretical convergence study in the multilevel setting. Moreover, we discuss its use combined with a band approximation and we compare the result with both τ and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz–FDE with variable coefficients. Finally, we investigate the use of a Riesz–Space FDE in a variational model for image deblurring, comparing the performance of specific preconditioning strategies.

Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H1 and Lp norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H1 and Lp norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems

Two-grid Method of Expanded Mixed Finite Element Approximations for Parabolic Integro-differential Optimal Control Problems

Two-Grid Methods for Maxwell’s Equations in a Cole-Cole Dispersive Medium

Two-Grid Methods for Maxwell’s Equations in a Cole-Cole Dispersive Medium

A two-grid characteristic block-centered finite difference method for Darcy-Forchheimer slightly compressible miscible displacement problem

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term |u|u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O(ε+Δt+H3+h2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time.

An Iterative Two-Grid Method for Strongly Nonlinear Elliptic Boundary Value Problems

An Iterative Two-Grid Method for Strongly Nonlinear Elliptic Boundary Value Problems

Application of two-grid interpolation to enhance average gradient method for solving partial differential equations

Previously presented method of calculating local average gradients for solving partial differential equations (PDEs) is enhanced by using interpolating gridpoints and triangular grids. The interpolating mesh provides finer computational grid, which is then used for solving the PDE. The combined use of the finer interpolating grid together with the original sparser grid is a two-grid method. By comparing the previous application of rectilinear grid for diffusion from initial point concentration to the new triangular two grid method, it was found that the application of triangular two-grid method improves stability of the solution and it provides more rapid convergence to the correct analytical solution.

A space-time second-order method based on modified two-grid algorithm with second-order backward difference formula for the extended Fisher–Kolmogorov equation

In this paper, a modified two-grid algorithm based on block-centred finite difference method is developed for the fourth-order nonlinear extended Fisher–Kolmogorov equation. To further improve the computational efficiency, an effective second-order accurate backward difference formula is considered. The modified two-grid method based on Newton iteration is constructed to linearize the nonlinear system. The method solves a miniature nonlinear system on a coarse grid accompanying a larger time step to get the numerical solution, then computes a linear system constructed by the previous result with the Taylor expansion on a fine grid accompanying a smaller time step to get the correct numerical solution. Theoretical analysis shows that the modified two-grid algorithm can achieve second-order convergence accuracy both in time and space domain. Several numerical experiments are provided to verify the theoretical result and the high efficiency of this approach. The practical problem illustrates the actual applicable value of the algorithm.

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>

Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

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.

Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations

Abstract In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L 2 L^{2} -norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.

Iterative two-grid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problem

Iterative two-grid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problem

Efficient algorithm based on two-grid method for semiconductor device problem

We study a three-step two-grid algorithm for solving semiconductor device equations that is discretized by the standard finite element method for the concentration equation and the mixed finite element method for the electric potential equation. The main procedures involve solving the original nonlinear equations on the coarse grid to obtain the initial approximate solution, as well as the linearized equations on the fine grid to obtain a modified solution; furthermore, the calculation on the fine grid is corrected. Error estimation for the method shows that the coarse grid can be much coarser than the fine grid, and the two-grid method still preserves asymptotically optimal approximation. Numerical experiments validate the reliability and effectiveness of the algorithm.

Symbol based convergence analysis in block multigrid methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied.However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.

Two-Grid Method for a Fully Discrete Mixed Finite Element Solution of the Time-Dependent Schrödinger Equation

We study the backward Euler fully discrete mixed finite element method for the time-dependent Schrödinger equation; the error result of the mixed finite element solution is obtained in the L2-norm with order O(τ+hk+1). Then, a two-grid method is presented with a backward Euler fully discrete scheme. Using this method, we solve the original problem on a much coarser grid and solve elliptic equations on a fine grid. In addition, the error of the two-grid solution is also obtained in the L2-norm with order O(τ+hk+1+Hk+2). The numerical experiment is provided to demonstrate the efficiency of the algorithm.