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The optimal strong convergence rates of the truncated EM and logarithmic truncated EM methods for multi-dimensional nonlinear stochastic differential equations

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arXiv 2025年

作者： Hu, Xingwei Dai, Xinjie Xiao, Aiguo School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China School of Mathematics and Statistics Yunnan University Yunnan Kunming650504 China

The truncated Euler–Maruyama (EM) method, developed by Mao (201.), is used to solve multi-dimensional nonlinear stochastic differential equations (SDEs). However, its convergence rate is suboptimal due to an unnecessary infinitesimal factor. The primary goal of this paper is to demonstrate the optimal convergence of the truncated EM method without infinitesimal factors. Besides, the logarithmic truncated EM method has not been studied in multi-dimensional cases, which is the other goal of this paper. We will show the optimal strong convergence order of the positivity-preserving logarithmic truncated EM method for solving multi-dimensional SDEs with positive solutions. Numerical examples are given to support our theoretical conclusions. Copyright © 2025, The Authors. All rights reserved.

关键词： Stochastic systems

High-Order Interior Penalty Finite Element Methods for Fourth-Order Phase-Field Models in Fracture Analysis

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arXiv 2025年

作者： Tian, Tian Chunyu, Chen Wei, Huayi School of Mathematics and Statistics Xiangtan University Hunan Xiangtan411105 China National Center of Applied Mathematics in Hunan Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan Xiangtan411105 China

This paper presents a novel approach for solving fourth-order phase-field models in brittle fracture mechanics using the Interior Penalty Finite Element Method (IP-FEM). The fourth-order model improves numerical stability and accuracy compared to traditional second-order phase-field models, particularly when simulating complex crack paths. The IP-FEM provides an efficient framework for discretizing these models, effectively handling nonconforming trial functions and complex boundary conditions. In this study, we leverage the FEALPy framework to implement a flexible computational tool that supports high-order IP-FEM discretizations. Our results show that as the polynomial order increases, the mesh dependence of the phase-field model decreases, offering improved accuracy and faster convergence. Additionally, we explore the trade-offs between computational cost and accuracy with varying polynomial orders and mesh sizes. The findings offer valuable insights for optimizing numerical simulations of brittle fracture in practical engineering applications. Copyright © 2025, The Authors. All rights reserved.

关键词： Brittle fracture

A LINEAR, UNCONDITIONALLY STABLE, SECOND ORDER DECOUPLED METHOD FOR THE NEMATIC LIQUID CRYSTAL FLOWS WITH SAV APPROACH

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arXiv 2025年

作者： Cao, Ruonan Yi, Nianyu School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China

In this paper, we present a second order, linear, fully decoupled, and unconditionally energy stable scheme for solving the Erickson-Leslie model. This approach integrates the pressure correction method with a scalar auxiliary variable technique. We rigorously demonstrate the unconditional energy stability of the proposed scheme. Furthermore, we present several numerical experiments to validate its convergence order, stability, and computational efficiency. © 2025, CC BY.

关键词： computational efficiency

HIGH ACCURACY TECHNIQUES BASED ADAPTIVE FINITE ELEMENT METHODS FOR ELLIPTIC PDES

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arXiv 2025年

作者： Xiao, Jingjing Liu, Ying Yi, Nianyu School of Mathematics Shandong University Jinan250100 China School of Science Xi’an University of Technology Xi’an710048 China School of Mathematics and Computational Science Xiangtan University Hunan National Center for Applied Mathematics Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan411105 China

This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection refinement has been well established, in practical computations, the use of non-asymptotic exact of error estimator and the excessive number of adaptive iteration steps often lead to inefficiency of the adaptive algorithm. We propose an efficient adaptive finite element method based on high-accuracy techniques including the superconvergence recovery technique and high-quality mesh optimization. The centroidal Voronoi Delaunay triangulation mesh optimization is embedded in the mesh adaption to provide high-quality mesh, and then assure that the superconvergence property of the recovered gradient and the asymptotical exactness of the error estimator. A tailored adaptive strategy, which could generate high-quality meshes with a target number of vertices, is developed to ensure the adaptive computation process terminated within 7 steps. The effectiveness and robustness of the adaptive algorithm is numerically demonstrated. Copyright © 2025, The Authors. All rights reserved.

关键词： Triangulation

NEURAL NETWORK-ENHANCED hr-ADAPTIVE FINITE ELEMENT ALGORITHM FOR PARABOLIC EQUATIONS

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arXiv 2025年

作者： Hao, Jiaxiong Huang, Yunqing Yi, Nianyu Yin, Peimeng Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China National Center for Applied Mathematics in Hunan Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China Department of Mathematical Sciences University of Texas at El Paso El PasoTX79968 United States

In this paper, we present a novel enhancement to the conventional hr-adaptive finite element methods for parabolic equations, integrating traditional h-adaptive and r-adaptive methods via neural networks. A major challenge in hr-adaptive finite element methods lies in projecting the previous step’s finite element solution onto the updated mesh. This projection depends on the new mesh and must be recomputed for each adaptive iteration. To address this, we introduce a neural network to construct a mesh-free surrogate of the previous step finite element solution. Since the neural network is mesh-free, it only requires training once per time step, with its parameters initialized using the optimizer from the previous time step. This approach effectively overcomes the interpolation challenges associated with non-nested meshes in computation, making node insertion and movement more convenient and efficient. The new algorithm also emphasizes SIZING and GENERATE, allowing each refinement to roughly double the number of mesh nodes of the previous iteration and then redistribute them to form a new mesh that effectively captures the singularities. It significantly reduces the time required for repeated refinement and achieves the desired accuracy in no more than seven space-adaptive iterations per time step. Numerical experiments confirm the efficiency of the proposed algorithm in capturing dynamic changes of *** Codes 92B20, 65M60, 35K57 © 2025, CC BY.

关键词： Interpolation

Efficient numerical methods for computing stationary states of spherical Landau-Brazovskii model

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arXiv 2025年

作者： Qiu, Qun Si, Wei Ji, Guanghua Jiang, Kai School of Mathematical Sciences Beijing Normal University Beijing100875 China School of Mathematics Sciences Peking University Beijing100871 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Hunan Xiangtan411105 China

In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time. Copyright © 2025, The Authors. All rights reserved.

关键词： Conjugate gradient method

ENHANCED GRADIENT RECOVERY-BASED A POSTERIORI ERROR ESTIMATOR AND ADAPTIVE FINITE ELEMENT METHOD FOR ELLIPTIC EQUATIONS

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arXiv 2025年

作者： Liu, Ying Xiao, Jingjing Yi, Nianyu Cao, Huihui Department of Mathematics School of Science Xi’an University of Technology Xi’an710000 China School of Mathematics Shandong University Jinan250100 China School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Hunan National Center for Applied Mathematics Xiangtan411105 China

Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their {1.ally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators, one can not establish upper and lower a posteriori error bounds for the classical recovery type error estimators without the saturation assumption. In this paper, we first present three examples to show the unsatisfactory performance in the practice of standard residual or recovery-type error estimators, then, an improved gradient recovery-based a posteriori error estimator is constructed. The proposed error estimator contains two parts, one is the difference between the direct and post-processed gradient approximations, and the other is the residual of the recovered gradient. The reliability and efficiency of the enhanced estimator are derived. Based on the improved recovery-based error estimator and the newest-vertex bisection refinement method with a tailored mark strategy, an adaptive finite element algorithm is designed. We then prove the convergence of the adaptive method by establishing the contraction of gradient error plus oscillation. Numerical experiments are provided to illustrate the asymptotic exactness of the new recovery-based a posteriori error estimator and the high efficiency of the corresponding adaptive algorithm. Copyright © 2025, The Authors. All rights reserved.

关键词： Finite element method

Conservative mixed Discontinuous Galerkin finite element method with full decoupling strategy for incompressible MHD problems

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Communications in Nonlinear science and Numerical {1. 2025年 147卷

作者： Huang, Huayi Huang, Yunqing Tang, Qili Yin, Lina Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China National Center for Applied Mathematics in Hunan Xiangtan University Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China

In this research, we propose a mixed Discontinuous Galerkin (DG) finite element method (FEM) for the numerical solution of incompressible magnetohydrodynamics (MHD) problems, ensuring the divergence-free condition for the velocity field. Our proposed methodology is characterized by its energy stability. Subsequently, we construct a novel finite element discretization scheme, for which we establish the theoretical framework of energy stability. Furthermore, we incorporate a preconditioning strategy that facilitates the development of an alternative positive definite, fully decoupled linearization approach. A significant advantage of this scheme is its ability to circumvent the complexities associated with solving saddle point problems at each time step. We demonstrate the uniqueness and convergence of the solutions within this fully decoupled, block-structured linear system. This theoretical validation is complemented by a series of numerical experiments that substantiate the effectiveness and accuracy of our proposed method. © 2025 Elsevier B.V.

关键词： Finite element method

Designing a minimal Landau model to stabilize desired quasicrystals

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Physical Review Research 2025年第2期7卷 023021-023021页

作者： Wei Si Shifeng Li Pingwen Zhang An-Chang Shi Kai Jiang School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China School of Mathematics and Statistics Wuhan University Wuhan 430072 China LMAM CAPT and School of Mathematical Sciences Peking University Beijing 100871 China Department of Physics and Astronomy McMaster University Hamilton L8S 4M1 Canada

Interparticle interactions with multiple length scales play a pivotal role in the formation and stability of quasicrystals. Choosing a minimal set of length scales to stabilize a given quasicrystal is a challenging problem. To address this challenge, we propose a symmetry-preserving screening method (SPSM) to design a Landau theory with a minimal number of length scales—referred to as the minimal Landau theory—that includes only the essential length scales necessary to stabilize quasicrystals. Based on a generalized multiple-length-scale Landau theory, SPSM first evaluates various spectral configurations of candidate structures under a hard constraint. It then identifies the configuration with the lowest free energy. Using this optimal configuration, SPSM calculates phase diagrams to explore the thermodynamic stability of desired quasicrystals. SPSM can design a minimal Landau theory capable of stabilizing the desired quasicrystals by incrementally increasing the number of length scales. Our application of SPSM has not only confirmed known behaviors in 1.- and 1.-fold quasicrystals but also led to a significant prediction that quasicrystals with 8-, 1.-, 1.-, and 1.-fold symmetry could be stable within three-length-scale Landau models.

关键词： Quasicrystals