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Convergence analysis of two-grid methods for second order hyperbolic equation

JOURNAL OF INEQUALITIES AND APPLICATIONS 2021年第1期2021卷 1-15页

作者： Wang, Keyan Wang, Qisheng Hengyang Normal Univ Sch Math & Stat Hengyang 421008 Hunan Peoples R China Wuyi Univ Sch Math & Computat Sci Jiangmen 529020 Guangdong Peoples R China

In this paper, a second-order hyperbolic equation is solved by a two-grid algorithm combined with the expanded mixed finite element method. The error estimate of the expanded mixed finite element method with discrete-time scheme is demonstrated. Moreover, we present a two-grid method and analyze its convergence. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h(1/2)). Finally, some numerical experiments are provided to illustrate the efficiency and accuracy of the proposed method.

关键词： Hyperbolic equation Expanded mixed finite element method two-grid algorithm Error estimate

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two-grid finite element method for the dual-permeability-Stokes fluid flow model

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NUMERICAL algorithmS 2021年第4期88卷 1703-1731页

作者： Nasu, Nasrin Jahan Al Mahbub, Md Abdullah Hussain, Shahid Zheng, Haibiao East China Normal Univ Sch Math Sci Shanghai 200241 Peoples R China East China Normal Univ Shanghai Key Lab Pure Math & Math Practice Shanghai 200241 Peoples R China Univ Chittagong Dept Accounting Chattogram Bangladesh Comilla Univ Dept Math Cumilla 3506 Bangladesh

In this paper, two-grid finite element method for the steady dual-permeability-Stokes fluid flow model is proposed and analyzed. Dual-permeability-Stokes interface system has vast applications in many areas such as hydrocarbon recovery process, especially in hydraulically fractured tight/shale oil/gas reservoirs. two-grid method is popular and convenient to solve a large multiphysics interface system by decoupling the coupled problem into several subproblems. Herein, the two-grid approach is used to reduce the coding task substantially, which provides computational flexibility without losing the approximate accuracy. Firstly, we solve a global problem through standard P-k - Pk- 1 - P-k - P-k finite elements on the coarse grid. After that, a coarse grid solution is applied for the decoupling between the interface terms and the mass exchange terms to solve three independent subproblems on the fine grid. The three independent parallel subproblems are the Stokes equations, the microfracture equations, and the matrix equations, respectively. Four numerical tests are presented to validate the numerical methods and illustrate the features of the dual-permeability-Stokes model.

关键词： Dual-permeability-Stokes system two-grid algorithm Numerical experiments Horizontal wellbore

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two-grid based sequential peridynamic analysis method for quasi-static crack propagation

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ENGINEERING FRACTURE MECHANICS 2022年 269卷 1页

作者： Jo, Gwanghyun Ha, Youn Doh Kunsan Natl Univ Dept Math Gunsan Jeonrabug do South Korea Kunsan Natl Univ Dept Naval Architecture & Ocean Engn Gunsan Jeonrabugdo South Korea

We introduce two-grid based sequential analysis algorithm for implicit peridynamics to find the quasi-static solutions for crack propagation problems. The sequential analysis of implicit peridynamic formulation is capable of studying quasi-static fracturing of brittle materials. The failure model in peridynamics, however, usually allows one to break a single bond between two nodes under a certain damage criterion in tracing equilibrium points, which results in the whole simulation quite lengthy and inefficient. In order to circumvent the efficiency issue, we propose the two-grid based sequential peridynamic analysis algorithm, in which a coarse grid is used to find the equilibrium path with sequential analysis and the converged solution is properly prolongated to the fine grid. The failure model is also implemented in the coarse grid and the computed damage information is properly mapped into the fine grid. In this way, the proposed two-grid based scheme incorporates breaking multiple bonds into one step of sequential analyses. We document the superior behavior of the proposed scheme with respect to the conventional solution procedures of implicit peridynamics in both solution accuracy and computational efficiency for quasi-static crack propagation.

关键词： Peridynamics Sequential analysis two-grid algorithm Quasi-static crack propagation Brittle fracture

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A two-grid method with backtracking for the mixedNavier-Stokes/Darcy model

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2020年第6期36卷 1601-1610页

作者： Du, Guangzhi Li, Qingtao Zhang, Yuhong Shandong Normal Univ Sch Math & Stat Jinan Peoples R China Hunan Normal Univ Sch Math & Stat Changsha 410081 Hunan Peoples R China

In this paper, we consider the effect of adding a coarse mesh correction to the two-grid algorithm for the mixed Navier-Stokes/Darcy model. The method yields bothL(2)andH(1)optimal velocity and piezometric head approximations and anL(2)optimal pressure approximation. The method involves solving one small, coupled, nonlinear coarse mesh problem, two independent subproblems (linear Navier-Stokes equation and Darcy equation) on the fine mesh, and a correction problem on the coarse mesh. Theoretical analysis and numerical tests are done to indicate the significance of this method.

关键词： backtracking technique Darcy's law finite element method Navier-stokes equations two-grid algorithm

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

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2020年第12期79卷 3378-3393页

作者： Yao, C. H. Li, F. R. Zhao, Y. M. Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China Chinese Acad Sci State Key Lab Space Weather Beijing 100190 Peoples R China Xuchang Univ Sch Math & Stat Xuchang 461000 Peoples R China

Based on two-grid algorithm, we develop the superconvergence analysis of fully-discrete scheme with the lowest-order Nedelec 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(Delta t(2) + h(2 )+ H-3 ), which means that the space mesh size should satisfy h = O(H-3/2). Finally, the numerical example is presented to verify the theoretical results. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Maxwell's equation Thermal effect Nedelec element two-grid algorithm Superconvergence

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Convergence of Multi-level algorithms for a Class of Nonlinear Problems

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JOURNAL OF SCIENTIFIC COMPUTING 2020年第2期84卷 34-34页

作者： Kim, Dongho Park, Eun-Jae Seo, Boyoon Yonsei Univ Univ Coll Seoul 03722 South Korea Yonsei Univ Dept Computat Sci & Engn Seoul 03722 South Korea Yonsei Univ Dept Math Seoul 03722 South Korea

In this article, we develop and analyze two-grid/multi-level algorithms via mesh refinement in the abstract framework of Brezzi, Rappaz, and Raviart for approximation of branches of nonsingular solutions. Optimal fine grid accuracy of two-grid/multi-level algorithms can be achieved via the proper scaling of relevant meshes. An important aspect of the proposed algorithms is the use of mesh refinement in conjunction with Newton-type methods for system solution in contrast to the usual Newton's method on a fixed mesh. The pseudostress-velocity formulation of the stationary, incompressible Navier-Stokes equations is considered as an application and the Raviart-Thomas mixed finite element spaces are used for the approximation. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

关键词： Nonlinear problems Multi-level Mesh refinement Error estimates two-grid algorithm Pseudostress-velocity Navier-Stokes equations

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LOCAL AND PARALLEL FINITE ELEMENT METHOD FOR THE MIXED NAVIER-STOKES/DARCY MODEL WITH BEAVERS-JOSEPH INTERFACE CONDITIONS

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Acta Mathematica Scientia 2017年第5期37卷 1331-1347页

作者：杜光芝左立云 School of Mathematics and Statistics Shandong Normal UniversityJinan 250014China School of Mathematical Sciences University of JinanJinan 250022China

In this paper, we consider the mixed Navier-Stokes/Darcy model with BeaversJoseph interface conditions. Based on two-grid discretizations, a local and parallel finite element algorithm for this mixed model is proposed and analyzed. Optimal errors are obtained and numerical experiments are presented to show the efficiency and effectiveness of the local and parallel finite element algorithm.

关键词： Navier-Stokes equations Darcy's law two-grid algorithm Beavers-Joseph interface conditions parallel finite element method

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A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

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Applied Mathematics(A Journal of Chinese Universities) 2011年第3期26卷 368-378页

作者： SHAO Xin-ping HAN Dan-fu Department of Mathematics Zhejiang University Hangzhou 310027 China

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

关键词： two-grid algorithm Navier-Stokes equations Stream function form Reynolds number Newton iteration.

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A modified local and parallel finite element method for the mixed Stokes-Darcy model

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2016年第2期435卷 1129-1145页

作者： Du, Guangzhi Hou, Yanten Zuo, Liyun Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Xian 710049 Shaanxi Peoples R China

Based on two-grid discretizations, a modified local and parallel finite element algorithm for the mixed Stokes-Darcy problem is proposed and analyzed in this paper. The idea is that, on the coarse grid, we solve a coupled Stokes-Darcy problem;on the fine grid, we first consider the discrete Darcy problem with a coarse grid approximation u(H) to the interface coupling conditions, then solve a series of independent local Stokes subproblems with the numerical solution of Darcy problem on fine grid to the interface coupling conditions in parallel. Optimal error estimates are obtained, and numerical experiments demonstrate the accuracy and efficiency of the modified local and parallel algorithm. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Stokes equations Darcy's law Decoupling two-grid algorithm Parallel finite element method

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Modified two-grid method for solving coupled Navier-Stokes/Darcy model based on Newton iteration

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Applied Mathematics(A Journal of Chinese Universities) 2015年第2期30卷 127-140页

作者： SHEN Yu-jing HAN Dan-fu SHAO Xin-ping Department of Mathematics Zhejiang University Department of Mathematics Hangzhou Normal University School of Science Hangzhou Dianzi University

A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is given, which shows that the same order of accuracy can be achieved as solving the system directly in the fine mesh when h = H2. Both theoretical analysis and numerical experiments illustrate the efficiency of the algorithm for solving the coupled problem.

关键词： Navier-Stokes equation Darcy＇s law interface coupling two-grid algorithm Newton iteration

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