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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 computationally 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

Two-Relaxation-Time Regularized Lattice Boltzmann Model for Navier-Stokes Equations

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SSRN

SSRN 2024年

作者： Yu, Yuan Qin, Zuojian Chen, Siwei Shu, Shi Yuan, Hai-Zhuan aSchool of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China bNational Center for Applied Mathematics in Hunan Xiangtan 411105 China cHunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 China

In this paper, we propose a novel two-relaxation-time regularized lattice Boltzmann (TRTRLB) model for simulating weakly compressible isothermal flows. A free relaxation parameter, τs,2, is employed to relax the regularized non-equilibrium third-order terms. ChapmanEnskog analysis reveals that our model can accurately recover the Navier-Stokes equations (NSEs). Theoretical analysis of the Poiseuille flow problem demonstrates that the slip velocity magnitude in the proposed model is controlled by a magic parameter, which can be entirely eliminated under specific values, consistent with the classical TRT model. Our simulations of the double shear layer problem, Taylor-Green vortex flow, and force-driven Poiseuille flow confirm that the stability and accuracy of our model significantly surpass those of both the regularized lattice Boltzmann (RLB) and two-relaxation-time (TRT) models, even under super-high Reynolds numbers as Re = 107. Concurrently, the TRT-RLB model exhibits superior performance in very high viscosity scenarios. The simulations of creeping flow around a square cylinder demonstrates the model’s capability to accurately compute ultra-low Reynolds numbers as Re = 10−7. This study establishes the TRT-RLB model as a flexible and robust tool in computational fluid dynamics. © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

An Accurate, Robust and Efficient Convection-Pressure Flux Splitting Scheme for Compressible Euler Flows

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SSRN

SSRN 2023年

作者： Hu, Lijun Tan, Shide Li, Long Yuan, Haizhuan School of Mathematics and Statistics Hengyang Normal University Hunan Hengyang421002 China School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan Xiangtan411105 China

The Roe's flux difference splitting scheme has been a popular shock-capturing method in computational fluid dynamics due to its high resolution for various discontinuities. However, it fails to satisfy the entropy condition and will encounter different forms of instability while calculating shock wave problems, such as the occurrence of post-shock oscillation in simulations of slowly moving shock waves and the carbuncle phenomenon in simulations of multidimensional strong shock waves. A simple flux difference splitting scheme based on the idea of convection-pressure splitting is proposed in the present work. The flux of the 3D Euler equations is split into the convection part and the pressure part with application of the Toro-Vazquez splitting procedure. The pressure subsystem has a complete set of linearly independent eigenvectors and thus it is solved by the conventional flux difference splitting method which is based on the eigenstructure of the system. The weakly hyperbolic convection subsystem, however, does not have a complete set of linearly independent eigenvectors and it is solved by the flux difference splitting scheme based on the generalized eigenvectors. In order to improve the resolution for contact discontinuities, a strategy combining the THINC reconstruction with the BVD algorithm is employed to minimize the density difference in the numerical dissipation term of the convection flux. Results of stability analysis show that the proposed scheme can effectively attenuate the numerical error and thus suppress the multidimensional shock instability. A comprehensive range of numerical simulations for classical 1D, 2D and 3D benchmark test problems demonstrate that the proposed scheme satisfies the entropy condition and positivity-preserving, possesses high resolution for contact discontinuities, and, more importantly, is free from the post-shock oscillation of slowly moving shock waves and the instability of multidimensional strong shock waves. © 2023, Th

关键词： Eigenvalues and eigenfunctions

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

A C0-continuous nonconforming virtual element method for linear strain gradient elasticity

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arXiv 2024年

作者： Huang, Jianguo Yu, Yue School of Mathematical Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China

A robust C0-continuous nonconforming virtual element method (VEM) is developed for a boundary value problem arising from strain gradient elasticity in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). The stability condition of the VEMs is derived by establishing Korn-type inequalities and inverse inequalities. Some crucial commutative relations for locking-free analysis as in elastic problems are derived. The sharp and uniform error estimates with respect to both the microscopic parameter and the Lamé coefficient are achieved in the lowest-order case, which is also verified by numerical *** Codes 65N12, 65N15, 65N22, 65N30 © 2024, CC BY-NC-ND.

关键词： Boundary value problems

Two-Relaxation-Time Regularized Lattice Boltzmann Model for Navier-Stokes Equations

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SSRN

SSRN 2024年

作者： Yu, Yuan Qin, Zuojian Gu, Qingqing Chen, Siwei Shu, Shi Yuan, Hai-Zhuan School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China National Center for Applied Mathematics in Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China School of Energy and Power Engineering University of Shanghai for Science and Technology Shanghai200093 China

In this paper, we develop a two-relaxation-time regularized lattice Boltzmann (TRT-RLB) model for simulating weakly compressible isothermal flows, which demonstrates superior stability and accuracy over existing models such as the regularized lattice Boltzmann (RLB) and two-relaxation-time (TRT) models. In this model, a free relaxation parameter, τs,2, is employed to relax the regularized non-equilibrium third-order terms. Chapman-Enskog analysis reveals that our model can accurately recover the Navier-Stokes equations. Theoretical analysis and numerical experiments both confirm the model’s ability to eliminate non-physical numerical slip associated with the half-way bounce-back scheme. Our simulations of the double shear layer problem and Taylor-Green vortex flow exhibit pronounced advantages in terms of stability and accuracy, even under super-high Reynolds numbers as high as Re = 107. Additionally, the simulation of creeping flow around a square cylinder showcases the model’s precision in computing ultra-low Reynolds numbers down to Re = 10−7. This robust capability confirms the proposed model as a highly effective and adaptable tool in computational fluid dynamics. © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

OPTIMAL L∞(L2) AND L1(L2) A POSTERIORI ERROR ESTIMATES FOR THE FULLY DISCRETE APPROXIMATIONS OF TIME FRACTIONAL PARABOLIC DIFFERENTIAL EQUATIONS

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arXiv 2023年

作者： Cao, Jiliang Wang, Wansheng Xiao, Aiguo Department of Mathematics Shanghai Normal University Shanghai200234 China 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

We derive optimal order a posteriori error estimates in the L∞(L2) and L1(L2)-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the L1 methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results. Copyright © 2023, The Authors. All rights reserved.

关键词： Finite element method

Error Analysis of Virtual Element Method for the Poisson-Boltzmann Equation

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arXiv 2023年

作者： Huang, Linghan Shu, Shi Yang, Ying Guangxi Guilin541004 China School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing Ministry of Education Hunan Xiangtan411105 China Guangxi541004 China

The Poisson-Boltzmann equation is a nonlinear elliptic equation with Dirac distribution sources, which has been widely applied to the prediction of electrostatics potential of biological biomolecular systems in solution. In this paper, we discuss and analysis the virtual element method for the Poisson-Boltzmann equation on general polyhedral meshes. Under the low regularity of the solution of the whole domain, nearly optimal error estimates in both L2-norm and H1-norm for the virtual element approximation are obtained. The numerical experiment on different polyhedral meshes shows the efficiency of the virtual element method and verifies the proposed theoretical prediction. Copyright © 2023, The Authors. All rights reserved.

关键词： Poisson equation

Two-relaxation-time regularized lattice Boltzmann model for convection-diffusion equation with variable coefficients

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arXiv 2023年

作者： Yu, Yuan Qin, Zuojian Yuan, Haizhuan Shu, Shi School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China National Center for Applied Mathematics in Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China

In this paper, a new two-relaxation-time regularized (TRT-R) lattice Boltzmann (LB) model for convection-diffusion equation (CDE) with variable coefficients is proposed. Within this framework, we first derive a TRT-R collision operator by constructing a new regularized procedure through the high-order Hermite expansion of non-equilibrium. Then a first-order discrete-velocity form of discrete source term is introduced to improve the accuracy of the source term. Finally and most importantly, a new first-order space-derivative auxiliary term is proposed to recover the correct CDE with variable coefficients. To evaluate this model, we simulate a classic benchmark problem of the rotating Gaussian pulse. The results show that our model has better accuracy, stability and convergence than other popular LB models, especially in the case of a large time step. Copyright © 2023, The Authors. All rights reserved.

关键词： Boltzmann equation