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Regularity and an adaptive finite element method for elliptic equations with Dirac sources on line cracks

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 462卷

作者： Cao, Huihui Li, Hengguang Yi, Nianyu Yin, Peimeng Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Peoples R China Wayne State Univ Dept Math Detroit MI 48202 USA Xiangtan Univ Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Peoples R China Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA

In this paper, we consider an adaptive finite element method for solving elliptic equations with line Dirac delta functions as the source term. Instead of using a local H-1 local indicator, or regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem. This equivalent problem is defined in the same domain as the original problem but features a zero source term and nonzero flux jumps along the line cracks, leading to a more regular solution. The a posteriori error estimator relies on meshes that conform to the line cracks, and its edge jump residual essentially incorporates the flux jumps of the transmission problem on these cracks. The proposed error estimator is proven to be both reliable and efficient. We also introduce an adaptive finite element algorithm based on this error estimator and the bisection refinement method. Numerical tests demonstrate that quasi- optimal convergence rates are achieved for both low-order and high-order approximations, with the associated adaptive meshes primarily refined at a finite number of singular points in the domain.

关键词： Line Dirac measure Transmission problem Regularity adaptive finite element method A posteriori error estimator

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An adaptive finite element method for the Helmholtz equation in irregular optical domains using recovery techniques

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JOURNAL OF OPTICS-INDIA 2025年 1-13页

作者： Abd El-Shafi, Gh. El-Kalla, I. L. Sameeh, M. Mansoura Univ Fac Engn Math & Engn Phys Dept Mansoura 35516 Egypt Mansoura Natl Univ Fac Engn Gamasa Egypt

This paper presents an adaptive finite element algorithm to address the challenges of solving the Helmholtz equation in complex optical geometries. The adaptivity loop is constructed using a posteriori error estimator, built by evaluating the recovered gradient based on the polynomial preserving recovery method. A description of the error estimation accompanies this methodology. A series of numerical tests are performed to prove the robustness of the adaptive finite element algorithm based on the proposed mesh refinement criteria. The results show a reduction in the computation cost and an improvement in the accuracy of the solution. The adaptive approach's notable speed in getting more accurate solutions enables it to deal with the complex problems of optical wave propagation.

关键词： Helmholtz equation adaptive finite element method Polynomial preserving recovery Irregular domain

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adaptive finite element method FOR THE MAXWELL EIGENVALUE PROBLEM

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2019年第1期57卷 478-494页

作者： Boffi, Daniele Gastaldi, Lucia Univ Pavia Dipartimento Matemat F Casorati Pavia Italy Aalto Univ Dept Math & Syst Anal Helsinki Finland Univ Brescia DICATAM Brescia Italy

In this paper we prove the optimal convergence of a standard adaptive scheme based on edge finite elements for the approximation of the solutions of the eigenvalue problem associated with Maxwell's equations. The ... 详细信息

关键词： edge finite elements eigenvalue problem Maxwell's equations adaptive finite element method

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adaptive finite element method for Shape Modification in Optimization of Electric Machines

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ELECTRICAL ENGINEERING IN JAPAN 2009年第1期168卷 21-28页

作者： Yamazaki, Katsumi Ishigami, Hiroki Chiba Inst Technol Chiba Japan

Ameshes based on adaptive meshing for optimization of electric machines is proposed. In the optimization of the machine, the proposed method generates the mesh for the new shape from the previous mesh with minor modification. The locations of the nodes for the new shape are decided by solving a Laplacian equation whose unknowns are the displacements of the nodes. The advantages are illustrated by application to the shape optimization of an IPM motor. (C) 2009 Wiley Periodicals, Inc. Electr Eng Jpn, 168(1): 21-28, 2009;Published online in Wiley InterScience (***). DOI 10.1002/eej.20773

关键词： adaptive finite element method optimization Rosenbrock's method IPM motor iron loss

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adaptive finite element method for an optimal control problem of Stokes flow with L2-norm state constraint

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN FLUIDS 2012年第3期69卷 534-549页

作者： Niu, Haifeng Yuan, Lei Yang, Danping E China Normal Univ Dept Math Shanghai 200241 Peoples R China Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China SW Forest Univ Dept Fundamental Courses Kunming 650224 Yunnan Peoples R China

An adaptive finite element approximation for an optimal control problem of the Stokes flow with an L2-norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual-type a posteriori error estimators of the H1-error of state and L2-error of control are given, which are suitable to carry out the adaptive multi-mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright (C) 2011 John Wiley & Sons, Ltd.

关键词： stokes flow optimal control state constraint adaptive finite element method a posteriori error estimate

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adaptive finite element method TO DETERMINE K_I AND K_II OF CRACK PLATE WITH DIFFERENT E_INCLUSION/E_PLATE RATIO

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TRANSACTIONS OF THE CANADIAN SOCIETY FOR MECHANICAL ENGINEERING 2011年第3期35卷 355-368页

作者： Limtrakarn, Wiroj Dechaumphai, Pramote Thammasat Univ Dept Mech Engn Pathum Thani Thailand Chulalongkorn Univ Dept Mech Engn Bangkok Thailand

An adaptive finite element method is presented to determine the K-I and K-II stress intensity factors of crack plate with different inclusions. The paper starts from describing two-dimensional fracture mechanics theory, an adaptive finite element formulation and the reflection photoelastic technique. An adaptive finite element method is evaluated by analyzing two examples. A single edge cracked plate made from polycarbonate. The second example is the slant edge 45 degrees cracked plate subjected to a uniform uniaxial tensile stress. The K-I and K-II results are found to be function of the crack length per width and the inverse function of E ratio. These examples demonstrate the efficiency of the adaptive finite element method to provide accurate solutions as compared to those from the reflection photoelastic technique.

关键词： adaptive finite element method stress intensity factor different E ratio

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adaptive finite element method for nonmonotone quasi-linear elliptic problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2021年 93卷 94-105页

作者： Guo, Liming Bi, Chunjia Xinyang Normal Univ Coll Math & Stat Xinyang 464000 Peoples R China Yantai Univ Sch Math & Informat Sci Yantai 264005 Peoples R China

In this paper, we study the simplest and the most standard adaptive finite element method for the secondorder nonmonotone quasi-linear elliptic problems with the exact solution u epsilon H-0(1+alpha) (Omega), alpha >= 1/2. The adaptive algorithm is based on the residual-based a posteriori error estimators and Dorfler's marking strategy. We prove the convergence and quasi-optimality of the adaptive finite element method when the initial mesh is sufficiently fine. Numerical experiments are provided to illustrate our findings.

关键词： adaptive finite element method Convergence Quasi-optimality Nonmonotone Quasi-linear elliptic problems

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adaptive finite element method for parabolic equations with Dirac measure

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2018年 328卷 217-241页

作者： Gong, Wei Liu, Huipo Yan, Ningning Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math NCMISLSEC Beijing 100190 Peoples R China Inst Appl Phys & Computat Math Beijing 100094 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci NCMISLSEC Beijing 100190 Peoples R China

In this paper we study the adaptive finite element method for parabolic equations with Dirac measure. Two kinds of problems with separate measure data in time and measure data in space are considered. It is well known that the solutions of such kind of problems may exhibit lower regularity due to the existence of the Dirac measure, and thus fit to adaptive FEM for space discretization and variable time steps for time discretization. For both cases we use piecewise linear and continuous finite elements for the space discretization and backward Euler scheme, or equivalently piecewise constant discontinuous Galerkin method, for the time discretization, the a posteriori error estimates based on energy and L-2 norms for the fully discrete problems are then derived to guide the adaptive procedure. Numerical results are provided at the end of the paper to support our theoretical findings. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Parabolic equation Dirac measure adaptive finite element method Space-time discretization A posteriori error estimates

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adaptive finite element method for an Elliptic Optimal Control Problem with Integral State Constraints

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COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION 2024年 1-40页

作者： Shakya, Pratibha Porwal, Kamana Indian Inst Technol Delhi Dept Math New Delhi 110016 India

In this article, we develop a posteriori error analysis of a nonconforming finite element method for a linear quadratic elliptic distributed optimal control problem with two different sets of constraints, namely (i) integral state constraint and integral control constraint;(ii) integral state constraint and pointwise control constraints. In the analysis, we have taken the approach of reducing the state-control constrained minimization problem into a state minimization problem obtained by eliminating the control variable. The reliability and efficiency of a posteriori error estimator are discussed. Numerical results are reported to illustrate the behavior of the error estimator.

关键词： Elliptic optimal control problem Fourth-order variational inequality Integral state constraints adaptive finite element method

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hp-adaptive finite element method for linear elasticity using higher-order virtual node method

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JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY 2015年第10期29卷 4299-4312页

作者： Oh, Hyeon Cheol Lee, Byung Chai Korea Adv Inst Sci & Technol Dept Mech Engn Taejon 305701 South Korea

Higher-order polygonal finite elements are developed for adaptive analyses of linear elastic problem. These elements are constructed using virtual node method based on partition of unity coupled with polynomial enrichment functions. Because the element shape functions are polynomials, the stiffness matrix is computed precisely with standard Gauss quadrature rules. Several numerical examples of linear elasticity are presented to validate the accuracy and convergence of the proposed elements. One of the advantages of the proposed elements is that they can be used as transition elements with hanging nodes on higher-order approximation meshes. Building on this advantage, h- and hp-adaptive finite element analyses of numerical examples with local singularities are performed on triangular quadtree meshes in order to demonstrate the performance of the adaptive strategies using the proposed elements.

关键词： adaptive finite element method Polygonal element Partition of unity Virtual node method Triangular quadtree mesh Hanging nodes

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