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Adaptive finite element method for phase field fracture models based on recovery error estimates

Journal of computational and Applied Mathematics 2026年 472卷

作者： Tian Tian Chunyu Chen Liang He Huayi Wei School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China National Center of Applied Mathematics in Hunan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 China

The phase-field model is a widely used mathematical approach for describing crack propagation in continuum damage fractures. In the context of phase field fracture simulations, adaptive finite element methods (AFEM) are often employed to address the mesh size dependency of the model. However, existing AFEM approaches for this application frequently rely on heuristic adjustments and empirical parameters for mesh refinement. In this paper, we introduce an adaptive finite element method based on a recovery type posteriori error estimates approach grounded in theoretical analysis. This method transforms the gradient of the numerical solution into a smoother function space, using the difference between the recovered gradient and the original numerical gradient as an error indicator for adaptive mesh refinement. This enables the automatic capture of crack propagation directions without the need for empirical parameters. We have implemented this adaptive method for the Hybrid formulation of the phase-field model using the open-source software package FEALPy. The accuracy and efficiency of the proposed approach are demonstrated through simulations of classical 2D and 3D brittle fracture examples, validating the robustness and effectiveness of our implementation.

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SomeWeighted Averaging Methods for Gradient Recovery

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Advances in Applied Mathematics and Mechanics 2012年第2期4卷 131-155页

作者： Yunqing Huang Kai Jiang Nianyu Yi Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105HunanChina

We propose some new weighted averaging methods for gradient recovery,and present analytical and numerical investigation on the performance of these weighted averaging *** is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in *** results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

关键词： Finite element method weighted averaging gradient recovery

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Isotropic Cloak Materials Design Based on the Numerical Optimization Method of the Inverse Medium Problem

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Advances in Applied Mathematics and Mechanics 2022年第2期14卷 442-468页

作者： Wei Yang Tiancheng Wang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan 411105China

In recent years,various kinds of cloak devices were designed by transforma-tion optics,but these cloak metamaterials are anisotropic and difﬁcult to *** this paper,we designed the isotropic cloak metamaterials based on the numeri-cal method of the optimization theory to the inverse medium *** method has universality,and it is not limited by the shape and type of the cloak *** isotropic material is easier to manufacture in practice than anisotropic material.A large number of numerical results show the effectiveness of the method.

关键词： Inverse medium problem isotropic cloak materials ﬁnite element simulations.

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First-order optimality condition of basis pursuit denoise problem

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Applied Mathematics and Mechanics(English Edition) 2014年第10期35卷 1345-1352页

作者：朱玮舒适成礼智 School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Department of Mathematics and Computational Science College of ScienceNational University of Defense Technology

A new first-order optimality condition for the basis pursuit denoise （BPDN） problem is derived. This condition provides a new approach to choose the penalty param- eters adaptively for a fixed point iteration algorithm. Meanwhile, the result is extended to matrix completion which is a new field on the heel of the compressed sensing. The numerical experiments of sparse vector recovery and low-rank matrix completion show validity of the theoretic results.

关键词： basis pursuit denoise （BPDN） fixed point iteration first-order optimality,matrix completion

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The Convergence of Euler-Maruyama Method of Nonlinear Variable-Order Fractional Stochastic Differential Equations

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Advances in Applied Mathematics and Mechanics 2023年第4期15卷 852-879页

作者： Shanshan Xu Lin Wang Wenqiang Wang School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the *** particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing ***,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.

关键词： Variable-order Caputo fractional derivative Stochastic differential equations Euler-Maruyama method convergence multiplicative noise

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Optimal Convergence Rate of q-Maruyama Method for StochasticVolterra Integro-Differential Equations with Riemann-Liouville Fractional Brownian Motion

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Advances in Applied Mathematics and Mechanics 2022年第1期14卷 202-217页

作者： Mengjie Wang Xinjie Dai Aiguo Xiao School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

This paper mainly considers the optimal convergence analysis of the q-Maruyama method for stochastic Volterra integro-differential equations(SVIDEs)driven by Riemann-Liouville fractional Brownian motion under the global Lipschitz and linear growth ***,based on the contraction mapping principle,we prove the well-posedness of the analytical solutions of the ***,we show that the q-Maruyama method for the SVIDEs can achieve strong first-order *** particular,when the q-Maruyama method degenerates to the explicit Euler-Maruyama method,our result improves the conclusion that the convergence rate is H+1/2,H∈(0,1/2)by Yang et al.,***.,383(2021),***,the numerical experiment verifies our theoretical results.

关键词： Stochastic Volterra integro-differential equations Riemann-Liouville fractional Brownian motion well-posedness strong convergence

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A Numerical Algorithm for the Caputo Tempered Fractional Advection-Diffusion Equation

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Communications on Applied Mathematics and computation 2021年第1期3卷 41-59页

作者： Wenhui Guan Xuenian Cao School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityXiangtan 411105Hunan ProvinceChina

By transforming the Caputo tempered fractional advection-diffusion equation into the Riemann–Liouville tempered fractional advection-diffusion equation,and then using the fractional-compact Grünwald–Letnikov tempered difference operator to approximate the Riemann–Liouville tempered fractional partial derivative,the fractional central difference operator to discritize the space Riesz fractional partial derivative,and the classical central difference formula to discretize the advection term,a numerical algorithm is constructed for solving the Caputo tempered fractional advection-diffusion *** stability and the convergence analysis of the numerical method are *** experiments show that the numerical method is effective.

关键词： Caputo tempered fractional advection-diffusion equation Fractional-compact Grünwald–Letnikov tempered Fractional central difference operator Stability Convergence

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Stability and Convergence of the Canonical Euler Splitting Method for Nonlinear Composite Stiff Functional Differential-Algebraic Equations

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Advances in Applied Mathematics and Mechanics 2022年第6期14卷 1276-1301页

作者： Hongliang Liu Yameng Zhang Haodong Li Shoufu Li School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

A novel canonical Euler splitting method is proposed for nonlinear compositestiff functional differential-algebraic equations, the stability and convergence of themethod is evidenced, theoretical results are further confirmed by some numerical ***, the numerical method and its theories can be applied to specialcases, such as delay differential-algebraic equations and integral differential-algebraicequations.

关键词： Canonical Euler splitting method nonlinear composite stiff functional differentialalgebraic equations stability convergence.

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Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

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Applied Mathematics and Mechanics(English Edition) 2022年第1期43卷 141-154页

作者： Qili TANG Yunqing HUANG Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan Province411105China

Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are *** approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution,and then locally solve linearized residual subproblems by one of three iterations(Stokes-type,Newton,and Oseen-type)on subdomains with fine grid in parallel to approximate the high frequency *** error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are *** numerical examples are implemented to verify the algorithm.

关键词： local and parallel algorithm finite element(FE)method iteration stationary incompressible magnetohydrodynamics(MHD)

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A UNIFIED A POSTERIORI ERROR ANALYSIS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF REACTIVE TRANSPORT EQUATIONS

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Journal of computational Mathematics 2006年第3期24卷 425-434页

作者： Ji-ming Yang Yan-ping Chen Hunan Key Laboratory for Computation and Simulation in Science and Engineering Institute for Computational and Applied Mathematics and School of Mathematics and Computing Science Xiangtan University Xiangtan 4 11105 China

Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-BabuSka-Baumann DG （OBB-DG）, non-symmetric interior penalty Galerkin （NIPG）, symmetric interior penalty Galerkin （SIPG）, and incomplete interior penalty Galerkin （IIPG）. A unified a posteriori residual-type error estimation is derived explicitly for these methods. From the computed solution and given data, explicit estimators can be computed efficiently and directly, which can be used as error indicators for adaptation. Unlike in the reference [10], we obtain the error estimators in L^2 （L^2） norm by using duality techniques instead of in L^2（H^1） norm.

关键词： A posteriori error estimates Duality techniques Discontinuous Galerkin methods

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