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A type of cascadic multigrid method for coupled semilinear elliptic equations

NUMERICAL ALGORITHMS 2020年第2期83卷 485-510页

作者： Xu, Fei Huang, Qiumei Beijing Univ Technol Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China

This paper is to introduce a type of cascadic multigrid method for coupled semilinear elliptic equations. Instead of solving the coupled semilinear elliptic equation on a very fine finite element space directly, the new scheme needs to solve a decoupled linear system by some smoothing steps on each of multilevel finite element spaces and solve a coupled semilinear elliptic equation on a coarse space. By choosing the appropriate number of smoothing steps on different finite element spaces, the corresponding optimal convergence rate and optimal computation work can be derived. Besides, the adaptive cascadic multigrid method for coupled semilinear elliptic equations and its analysis are also presented theoretically and numerically. Moreover, the requirement of bounded second order derivatives of the nonlinear term in the existing multigrid methods is reduced to the Lipschitz continuation property in the presented new scheme. Some numerical experiments are presented to validate our theoretical analysis.

关键词： Coupled semilinear elliptic equation Finite element method cascadic multigrid method Adaptive multigrid method

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A CELL-CENTERED multigrid SOLVER FOR THE FINITE VOLUME DISCRETIZATION OF ANISOTROPIC ELLIPTIC INTERFACE PROBLEMS ON IRREGULAR DOMAINS

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Journal of Computational Mathematics 2025年第1期43卷 18-42页

作者： Kejia Pan Xiaoxin Wu Hongling Hu Zhilin Li School of Mathematics and Statistics HNP-LAMACentral South UniversityChangsha 410083China Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) School of Mathematics and StatisticsHunan Normal UniversityChangsha 410081China Department of Mathematics North Carolina State UniversityRaleighNC 27695-8205USA

The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.

关键词： Elliptic interface problem Discontinuous coefficients Anisotropic coefficients cascadic multigrid method Richardson extrapolation

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A cascadic multigrid Algorithm for the Mortar Element method for Semiliner Ellptic Problems

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嘉应学院学报 2015年第11期33卷 5-10页

作者：邹战勇 School of mathematics and statistics Guangdong university of finance & economics

In this paper a cascadic multigrid algorithm for the mortar finite element approximation of the semilinear elliptic problem is proposed,and corresponding theorems aregiven,which display the error estimate and the comp... 详细信息

关键词： Mortar finite element cascadic multigrid method Senilinear elliptic problems

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cascadic multigrid methods for parabolic problems

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Science China Mathematics 2008年第8期51卷 1415-1439页

作者： DU Qiang MING PingBing Department of Mathematics Pennsylvania State UniversityUniversity ParkPA 16802USA LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesBeijing 100190China

In this paper,we consider the cascadic multigrid method for a parabolic type *** Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the *** new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular *** error bound sare derived for both smooth and non-smooth *** strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.

关键词： cascadic multigrid method parabolic problem finite element methods backward Euler scheme smoother stability optimal error order optimal complexity 65N30 65N55 65F10

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A cascadic ADAPTIVE FINITE ELEMENT method FOR NONLINEAR EIGENVALUE PROBLEMS IN QUANTUM PHYSICS

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MULTISCALE MODELING & SIMULATION 2020年第1期18卷 198-220页

作者： Xu, Fei Beijing Univ Technol Coll Appl Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China

This paper introduces a cascadic adaptive finite element method for nonlinear eigenvalue equations arising from quantum physics following the multilevel correction strategy. Instead of the classical scheme, which requires solving nonlinear eigenvalue equations on a series of adaptive spaces directly, the new scheme consists of several smoothing processes on an adaptive space sequence and nonlinear eigenvalue equations being solved in a very low dimensional space. The main feature of the proposed scheme is that large-scale nonlinear eigenvalue problem solving is avoided, and the associated smoothing process can be executed efficiently based on the appropriate number of smoothing steps. Thus, efficiency can be enhanced by the proposed cascadic adaptive method. The good performance of the new finite element strategy is examined by various numerical experiments.

关键词： nonlinear eigenvalue problem cascadic multigrid method adaptive finite element method

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cascadic meshfree method for the elliptic Monge-Ampere equation

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2013年第7-8期37卷 990-996页

作者： Liu, Zhiyong He, Yinnian Xi An Jiao Tong Univ Sch Math & Stat Ctr Computat Geosci Xian 710049 Peoples R China

The elliptic Monge-Ampere equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing and others. The numerical solution of the elliptic Monge-Ampere equation has been a subject of increasing interest recently. In this paper, we design a cascadic algorithm which is meshfree. We first generate hierarchical scattered data sets. Then on each successive refinement levels, the Monge-Ampere equation can be solved by Kansas method. We call this method as cascadic meshfree method (CMF). Different from cascadic multigrid method, CMF avoids tedious interpolation and is more easy for implementation and coding. Finally, numerical experiments confirm the efficiency and robustness of CMF method. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： cascadic meshfree method cascadic multigrid method Kansa's method Monge-Ampere equation Radial basis functions

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cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年第0期372卷 112720-000页

作者： Xu, Fei Huang, Qiumei Beijing Univ Technol Coll Appl Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China

In this paper, a type of cascadic adaptive finite element method for nonlinear eigenvalue problem is proposed based on the complementary approach. In this new scheme, we only need to do some smoothing steps for the linearized boundary value problems on a series of adaptive finite element spaces and solve some nonlinear eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the nonlinear eigenvalue problems on each adaptive space directly. Besides, the complementary error estimate for nonlinear eigenvalue problem will be introduced in our paper. This estimate can not only provide an accurate error estimate for the nonlinear eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Nonlinear eigenvalue problem Finite element method cascadic multigrid method Adaptive method Complementary method

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Error estimates of the lumped mass finite element method for semilinear elliptic problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2012年第7期236卷 1993-2004页

作者： Zeng, Jin-ping Yu, Hai-xiong Nanchang Inst Technol Dept Sci Nanchang 330099 Jiangxi Peoples R China Dongguan Univ Technol Coll Comp Dongguan 523808 Guangdong Peoples R China

We are concerned with the semilinear elliptic problems. We first investigate the L-2-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Semilinear elliptic problem Lumped mass discretization L-2-error estimate cascadic multigrid method

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An asymptotically exact a posteriori error estimator for non-selfadjoint Steklov eigenvalue problem

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APPLIED NUMERICAL MATHEMATICS 2020年 156卷 210-227页

作者： Xu, Fei Yue, Meiling Huang, Qiumei Ma, Hongkun Beijing Univ Technol Coll Appl Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Beijing Technol & Business Univ Sch Sci Beijing 100048 Peoples R China Sun Yat Sen Univ Business Sch Guangzhou 510275 Guangdong Peoples R China Zhuhai Huafa Investment Holdings Co Ltd Zhuhai 519031 Guangdong Peoples R China

This paper aims to introduce an asymptotically exact a posteriori error estimator for non-selfadjoint Steklov eigenvalue problem arising from inverse scattering by using the complementary technique, which provides an asymptotically exact estimate for eigenpair of non-selfadjoint Steklov eigenvalue problem. Besides, as its applications, we design a novel cascadic adaptive method for non-selfadjoint Steklov eigenvalue problem based on the asymptotically exact estimate. In our novel algorithm, we will transform the non-selfadjoint Steklov eigenvalue problem into some boundary value problems on the adaptive space sequence and some non-selfadjoint Steklov eigenvalue problem on a low dimensional finite element space. The involved boundary value problems are solved by executing some smoothing steps which is the key point of cascadic algorithm. The mesh refinement strategy and the number of smoothing steps for the cascadic adaptive method will be controlled by the proposed asymptotically exact a posteriori error estimator. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Non-selfadjoint Steklov eigenvalue problem Asymptotically exact a posteriori error estimator cascadic multigrid method Adaptive method Complementary method

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An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications

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Science China Mathematics 2021年第3期64卷 623-638页

作者： Fei Xu Qiumei Huang Beijing Institute for Scientific and Engineering Computing Faculty of ScienceBeijing University of TechnologyBeijing 100124China

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine meshes and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

关键词： Steklov eigenvalue problem a posteriori error estimator cascadic multigrid method adaptive finite element method complementary method

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