Two-grid Algorithm Research Articles

Two-grid algorithm for a system of singularly perturbed reaction-diffusion equations on Shishkin mesh

A system of two coupled singularly perturbed reaction-diffusion equations is considered. A cascadic two-grid algorithm of high accuracy based on a uniform with respect to small parameters convergent difference scheme for such problem has been developed. To increase the accuracy of the difference solution by an order, the Richardson extrapolation method was applied. The results of some numerical experiments are discussed.

Superconvergence analysis of two-grid FEM for Maxwell’s equations with a thermal effect

Based on two-grid algorithm, we develop the superconvergence analysis of fully-discrete scheme with the lowest-order Nédélec element and Crank–Nicolson scheme for the magneto-heat coupling model, which is also considered as Maxwell’s equations with a thermal effect. Then, main process of numerical analysis has two parts: On one hand, we utilize Newton-type Taylor expansion on superconvergent solutions for those nonlinear terms on coarse mesh, differing from the numerical solution, which makes our given two-grid method successful. On the other hand, we prove the solutions on the fine mesh to be higher accuracy by the postprocessing interpolation technique. Such a design is conducive to improving the computational accuracy in space and decreasing time consumption simultaneously. By employing the skill, we can obtain the convergent rate of O(Δt2+h2+H3), which means that the space mesh size should satisfy h=O(H32). Finally, the numerical example is presented to verify the theoretical results.

A characteristic finite element two-grid algorithm for a compressible miscible displacement problem

A characteristic finite element two-grid algorithm for a compressible miscible displacement problem

Time two-grid algorithm based on finite difference method for two-dimensional nonlinear fractional evolution equations

In this paper, a time two-grid finite difference (FD) algorithm is proposed for solving two-dimensional nonlinear fractional evolution equations. In this time two-grid FD algorithm, a nonlinear FD system is solved on the time coarse grid of size τC. And the Lagrange's linear interpolation formula is applied to provide some useful values for the time fine grid of size τF. Then, a linear system is solved on the time fine grid. In the temporal direction, a backward Euler method is employed for the time derivative and a first order convolution quadrature rule is applied to discretize the fractional integral term. And the second order central difference quotient is considered for the spatial approximation. By means of the discrete energy method, we obtain the unconditional discrete L2 stability and convergence of order O(τC2+τF+hx2+hy2), where hx and hy are the spatial step sizes in the x direction and the y direction, respectively. Numerical examples are presented to show the feasibility and efficiency of the time two-grid FD algorithm.

A two-grid method for semi-linear elliptic interface problems by partially penalized immersed finite element methods

In this paper, we present a two-grid partially penalized immersed finite element (IFE) scheme for the approximation of semi-linear elliptic interface problems. Extra stabilization terms are introduced at interface edges for penalizing the discontinuity in IFE functions. Optimal error estimates in both H1 and Lp norms are obtained for IFE discretizations. To linearize the IFE equations, two-grid algorithm based on some Newton iteration approach is applied. 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∕4). As a result, solving such a large class of non-linear equation will not be much more difficult than solving one single linearized equation.

A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model

In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of L1 discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations; second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence in L2-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system.

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).

A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations

We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The computational system is decoupled to smaller systems by using coarse space solutions at each time level, which can speed up the solution process compared with the finite element method combined with the Gummel iteration. We derive the optimal error estimates in $L^2$ norm for both semi- and fully discrete finite element approximations. Based on the a priori error estimates, the error estimates in $H^1$ norm are presented for the two-grid algorithm. The theoretical results indicate this decoupling method can retain the same accuracy as the finite element method. Numerical experiments including the Poisson-Nernst-Planck equations for an ion channel show the efficiency and effectiveness of the decoupling two-grid method.

Two-grid algorithm for the solution of singularly perturbed two-parameter problem on Shishkin mesh

A boundary value problem for a second-order semilinear singularly perturbed ordinary differential equation with two small parameters affecting the convection and diffusion terms is considered. We use Newton and Picard iterations for a linearization. To solve the problem at each iteration we apply the second order difference scheme on the Shishkin mesh which converges uniformly with respect to both singular perturbation parameters. To decrease the required number of arithmetical operations for resolving the difference scheme, a cascadic two-grid method is proposed. To increase the accuracy of difference scheme, we investigate the possibility to apply Richardson extrapolation using known solutions of the difference scheme on both meshes. The results of some numerical experiments are discussed.

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.

Two-grid economical algorithms for parabolic integro-differential equations with nonlinear memory

In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H=O(hr−1r). Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.

BLACK-BOX SOLVER FOR NUMERICAL SIMULATIONS AND MATHEMATICAL MODELLING IN ENGINEERING PHYSICS

This article presents a two-grid approach for developing a black-box iterative solver for a large class of real-life problems in continuum mechanics (heat and mass transfer, fluid dynamics, elasticity, electromagnetism, and others). The main requirements on this (non-)linear blackbox solver are: (1) robustness (the lowest number of problem-dependent components), (2) efficiency (close-tooptimal algorithmic complexity), and (3) parallelism (a parallel robust algorithm should be faster than the fastest sequential one). The basic idea is to use the auxiliary structured grid for more computational work, where (non-)linear problems are simpler to solve and to parallelize, i.e., to combine the advantages of unstructured and structured grids: simplicity of generation in complex domain geometry and opportunity to solve (non-)linear (initial-)boundary value problems by using the Robust Multigrid Technique. Topics covered include the description of the twogrid algorithm and estimation of their robustness, convergence, algorithmic complexity, and parallelism. Further development of modern software for solving real-life problems justifies relevance of the research. The proposed two-grid algorithm can be used in blackbox parallel software for the reduction in the execution time in solving (initial- )boundary value problems.

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.

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.

Numerical solution of a miscible displacement problem with dispersion term using a two-grid mixed finite element approach

A nonlinear system of two coupled partial differential equations is derived to describe miscible displacement of one incompressible fluid by another in a porous medium. A mixed finite element method (MFEM) is applied to the pressure equation, and the mixed finite element method with characteristics(CMFEM) is proposed to solve the concentration equation. A two-grid algorithm is considered to linearize nonlinear system of two coupled partial differential equations models. Moreover, the error estimates are carried out for the two-grid algorithm in this paper. Numerical experiments are presented to show the efficiency of the two-grid algorithm.

Two-grid finite element method and its a posteriori error estimates for a nonmonotone quasilinear elliptic problem under minimal regularity of data

In this paper, we propose and analyze a two-grid finite element method for a class of quasilinear elliptic problems under minimal regularity of data in a bounded convex polygonal Ω⊂R2, which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasilinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasilinear elliptic problem on a coarse space. Convergence estimates in the H1-norm are derived to justify the efficiency of the proposed two-grid algorithm. Moreover, we propose a natural and computationally efficient residual-based a posteriori error estimator of the two-grid finite element method for this nonmonotone quasilinear elliptic problem and derive the global upper and lower bounds on the error in the H1-norm. Numerical experiments are provided to confirm our theoretical findings.

Formula omitted]-discontinuous Galerkin methods for a wind-driven ocean circulation model: Two-grid algorithm

This paper presents a nonconforming finite element method for a streamfunction formulation of the stationary quasi-geostrophic equations, which describe the large scale wind-driven ocean circulation. The streamfunction formulation is a fourth order nonlinear PDE and the nonconforming method is based on C0-elements instead of C1-elements. Existence and uniqueness of the approximation are proved and optimal error estimates in several norms of interest are demonstrated under a small data assumption. Two-grid algorithms based on Picard and Newton type linearizations are then presented to efficiently resolve nonlinearities and computational results are given to demonstrate the efficiency of the algorithm. The Mediterranean sea example is tested with real world coastline data, which illustrates the effectiveness of the two-grid approach in the wind-driven ocean circulation simulation.

A Two-Grid Block-Centered Finite Difference Algorithm for Nonlinear Compressible Darcy–Forchheimer Model in Porous Media

In this paper, a block-centered finite difference method is proposed to discretize the compressible Darcy–Forchheimer model which describes the high speed non-Darcy flow in porous media. The discretized nonlinear problem on the fine grid is solved by a two-grid algorithm in two steps: first solving a small nonlinear system on the coarse grid, and then solving a nonlinear problem on the fine grid. On the coarse grid, the coupled term of pressure and velocity is approximated by using the fewest number of node values to construct a nonlinear block-centered finite difference scheme. On the fine grid, the original nonlinear term is modified with a small parameter $$\varepsilon $$ to construct a linear block-centered finite difference scheme. Optimal order error estimates for pressure and velocity are obtained in discrete $$l^\infty (L^2)$$ and $$l^2(L^2)$$ norms, respectively. The two-grid block-centered finite difference scheme is proved to be unconditionally convergent without any time step restriction. Some numerical examples are given to testify the accuracy of the proposed method. The numbers of iterations are reported to illustrate the efficiency of the two-grid algorithm.

A two-grid penalty method for American options

In this paper we consider the pricing of American options, governed by a partial differential complementarity problem. The differential problem is first approximated by a semi-linear PDE using two distinct penalty approaches which are well known in computational finance. We then initiate the two-grid algorithm by solving the nonlinear problem on a coarse grid and further the linearized in the interpolated coarse-grid solution problem on a fine grid. By means of the maximum principle the algorithm is shown to be of fourth order convergence rate in space. Numerical experiments verify the presented two-grid approach where we draw some interesting conclusions.

The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis

ABSTRACTThe two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277–2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier–Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme.