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$hp$-Version Analysis for Arbitrarily Shaped Elements on the Boundary Discontinuous Galerkin Method for Stokes Systems 边界上任意形状元素的 $hp$ 版本分析 斯托克斯系统的连续伽勒金方法
IF 1.1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-06-01 DOI: 10.4208/ijnam2024-1021
Efthymios N. Karatzas
In the present work, we examine and analyze an $hp$-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system oncomputational meshes consisting of polytopic elements on the boundary. This approach is basedon the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as hasbeen introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and $H^1/L^2$-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluidsystem enabling the proof of the inf/sup condition and the $hp$- a priori error estimates, while weinvestigate the optimal convergence rates numerically. This approach recovers and integrates theflexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometricaldeformations are taking place by degenerating the edges, facets, of the polytopic elements onlyon the boundary, combined with the efficiency of the $hp$-version techniques based on arbitrarilyshaped elements without requiring any mapping from a given reference frame.
在本研究中,我们研究并分析了一种 $hp$ 版本的内部惩罚非连续 Galerkin 有限元方法,该方法用于在由边界上多点元素组成的计算网格上对稳定流体系统进行数值逼近。这种方法以非连续 Galerkin 方法为基础,并通过任意形状元素技术加以丰富,如文献 [13] 所介绍的那样。在此框架下,我们利用痕量、马尔可夫型和 $H^1/L^2$ 型逆估计的扩展,对任意形状的元素进行了研究,并考察了静止斯托克斯流体系统,从而证明了 inf/sup 条件和 $hp$- 先验误差估计,同时对最佳收敛速率进行了数值研究。这种方法恢复并整合了流体非连续 Galerkin 方法的灵活性和优越性,在发生几何变形时,只需在边界上退化多面体元素的边缘、面,同时结合基于任意形状元素的 $hp$ 版本技术的效率,而无需从给定参考框架进行任何映射。
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引用次数: 0
A Stabilizer-Free Weak Galerkin Finite Element Method for the Darcy-Stokes Equations 达西-斯托克斯方程的无稳定器弱 Galerkin 有限元方法
IF 1.1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-06-01 DOI: 10.4208/ijnam2024-1018
Kai He,Junjie Chen,Li Zhang, Maohua Ran
In this paper, we propose a new method for the Darcy-Stokes equations based onthe stabilizer-free weak Galerkin finite element method. In the proposed method, we remove thestabilizer term by increasing the degree of polynomial approximation space of the weak gradientoperator. Compared with the classical weak Galerkin finite element method, it will not increasethe size of global stiffness matrix. We show that the new algorithm not only has a simpler formula,but also reduces the computational complexity. Optimal order error estimates are established forthe corresponding numerical approximation in various norms. Finally, we numerically illustratethe accuracy and convergence of this method.
本文提出了一种基于无稳定项弱 Galerkin 有限元法的达西-斯托克斯方程新方法。在所提出的方法中,我们通过增加弱梯度算子多项式近似空间的度数来去除稳定项。与经典的弱 Galerkin 有限元方法相比,它不会增加全局刚度矩阵的大小。我们证明,新算法不仅公式更简单,而且降低了计算复杂度。我们为相应的数值近似建立了各种规范下的最优阶误差估计。最后,我们用数值说明了该方法的准确性和收敛性。
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引用次数: 0
A Voigt-Regularization of the Thermally Coupled Inviscid, Resistive Magnetohydrodynamic 热耦合无粘性、电阻磁流体动力学的 Voigt 规则化
IF 1.1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-06-01 DOI: 10.4208/ijnam2024-1019
Xingwei Yang,Pengzhan Huang, Yinnian He
In this paper, we prove the existence of weak solution and the uniqueness of strongsolution to a Voigt-regularization of the three-dimensional thermally coupled inviscid, resistiveMHD equations. We also propose a fully discrete scheme for the considered problem, which isproven to be stable and convergent. All computational results support the theoretical analysisand demonstrate the effectiveness of the presented scheme.
在本文中,我们证明了三维热耦合无粘性、阻力 MHD 方程的 Voigt 规则化弱解的存在性和强解的唯一性。我们还为所考虑的问题提出了一个完全离散的方案,该方案被证明是稳定和收敛的。所有计算结果都支持理论分析,并证明了所提出方案的有效性。
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引用次数: 0
Mixed Virtual Element Method for Linear Parabolic Integro-Differential Equations 线性抛物线积分微分方程的混合虚拟元素法
IF 1.1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-06-01 DOI: 10.4208/ijnam2024-1020
Meghana Suthar, Sangita Yadav
This article develops and analyses a mixed virtual element scheme for the spatialdiscretization of linear parabolic integro-differential equations (PIDEs) combined with backwardEuler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projectionsignificantly helps in managing the integral terms, yielding optimal convergence of order $O(h^{k+1})$ for the two unknowns $p(x, t)$ and $sigma(x, t).$ In addition, a step-by-step analysis is proposed for thesuper convergence of the discrete solution of order $O(h^{k+2}).$ The fully discrete case has also beenanalyzed and discussed to achieve $O(tau)$ in time. Several computational experiments are discussedto validate the proposed schemes computational efficiency and support the theoretical conclusions.
本文针对线性抛物线积分微分方程(PIDE)的空间离散化,结合后向尤勒时间离散化方法,开发并分析了一种混合虚拟元素方案。混合 Ritz-Volterra 投影的引入极大地帮助了积分项的管理,为两个未知数 $p(x, t)$ 和 $sigma(x, t) $ 带来了阶数为 $O(h^{k+1})$的最优收敛。此外,还为阶数为 $O(h^{k+2})$的离散解的超收敛提出了逐步分析。讨论了几个计算实验,以验证所提方案的计算效率,并支持理论结论。
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引用次数: 0
Dynamics Analysis of HIV-1 Infection Model with CTL Immune Response and Delays 带有 CTL 免疫反应和延迟的 HIV-1 感染模型的动力学分析
IF 1.1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-06-01 DOI: 10.4208/ijnam2024-1022
Ting Guo, Fei Zhao
In this paper, we rigorously analyze an HIV-1 infection model with CTL immuneresponse and three time delays which represent the latent period, virus production period andimmune response delay, respectively. We begin this model with proving the positivity and boundedness of the solution. For this model, the basic reproduction number $R_0$ and the immunereproduction number $R_1$ are identified. Moreover, we have shown that the model has three equilibria, namely the infection-free equilibrium $E_0,$ the infectious equilibrium without immuneresponse $E_1$ and the infectious equilibrium with immune response $E_2.$ By applying fluctuationlemma and Lyapunov functionals, we have demonstrated that the global stability of $E_0$ and $E_1$ are only related to $R_0$ and $R_1.$ The local stability of the third equilibrium is obtained under foursituations. Further, we give the conditions for the existence of Hopf bifurcation. Finally, somenumerical simulations are carried out for illustrating the theoretical results.
在本文中,我们严格分析了一个具有 CTL 免疫反应和三个时间延迟(分别代表潜伏期、病毒产生期和免疫反应延迟)的 HIV-1 感染模型。我们首先证明了该模型解的实在性和有界性。对于该模型,基本繁殖数 $R_0$ 和免疫产生数 $R_1$已经确定。此外,我们还证明了该模型有三个平衡态,即无感染平衡态 $E_0、无免疫反应的感染平衡态 $E_1$ 和有免疫反应的感染平衡态 $E_2$。通过应用波动雷玛和李亚普诺夫函数,我们证明了 $E_0$ 和 $E_1$ 的全局稳定性只与 $R_0$ 和 $R_1$ 有关。此外,我们还给出了霍普夫分岔存在的条件。最后,我们进行了一些数值模拟来说明理论结果。
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引用次数: 0
The Weak Galerkin Finite Element Method for the Dual-Porosity-Stokes Model 双孔隙-斯托克斯模型的弱伽勒金有限元法
IF 1.1 4区 数学 Q1 MATHEMATICS Pub Date : 2024-06-01 DOI: 10.4208/ijnam2024-1023
Lin Yang,Wei Mu,Hui Peng, Xiuli Wang
In this paper, we introduce a weak Galerkin finite element method for the dual-porosity-Stokes model. The dual-porosity-Stokes model couples the dual-porosity equations withthe Stokes equations through four interface conditions. In this method, we define several weakGalerkin finite element spaces and weak differential operators. We provide the weak Galerkinscheme for the model, and establish the well-posedness of the numerical scheme. The optimalconvergence orders of errors in the energy norm are derived. Finally, we verify the effectiveness ofthe numerical method with different weak Galerkin elements on different meshes.
本文介绍了双孔隙-斯托克斯模型的弱 Galerkin 有限元方法。双孔隙-斯托克斯模型通过四个界面条件将双孔隙方程与斯托克斯方程耦合在一起。在该方法中,我们定义了几个弱 Galerkins 有限元空间和弱微分算子。我们为模型提供了弱伽勒金方案,并建立了数值方案的良好拟合性。推导了能量规范误差的最佳收敛阶数。最后,我们验证了在不同网格上使用不同弱 Galerkin 元素的数值方法的有效性。
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引用次数: 0
A Direct Method for Solving Three-Dimensional Elliptic Interface Problems 解决三维椭圆界面问题的直接方法
IF 1.1 4区 数学 Q2 Mathematics Pub Date : 2024-05-01 DOI: 10.4208/ijnam2024-1014
Kumudu Gamage,Yan Peng, Zhilin Li
This paper presents a direct method for efficiently solving three-dimensional ellipticinterface problems featuring piecewise constant coefficients with a finite jump across the interface.A key advantage of our approach lies in its avoidance of augmented variables, distinguishingit from traditional methods. The computational framework relies on a finite difference schemeimplemented on a uniform Cartesian grid system. By utilizing a seven-point Laplacian for gridpoints away from the interface, our method only requires coefficient modifications for grid pointslocated near or on the interface. Numerical experiments validate our method’s effectiveness.Generally, it achieves second-order accuracy for both the solution and its gradient, measured inthe maximum norm, particularly effective in scenarios with moderate coefficient jumps. Extendingand building upon the recent work of [1] on 1D and 2D elliptic interfaces, our approach successfullyintroduces a simpler method for extension into three dimensions. Notably, our proposed methodnot only offers efficiency and accuracy but also enhances the simplicity of implementation, makingit accessible to non-experts in the field.
本文提出了一种直接方法,用于高效求解具有片断常数系数的三维椭圆界面问题,该问题具有跨界面的有限跳跃。计算框架依赖于在统一笛卡尔网格系统上实施的有限差分方案。通过对远离界面的网格点使用七点拉普拉斯,我们的方法只需要对位于界面附近或界面上的网格点进行系数修改。数值实验验证了我们的方法的有效性。一般来说,我们的方法在求解及其梯度方面都能达到二阶精度(以最大规范衡量),尤其是在系数跳变适中的情况下非常有效。在 [1] 最近关于一维和二维椭圆界面的研究基础上,我们的方法成功地引入了一种更简单的方法,并将其扩展到三维空间。值得注意的是,我们提出的方法不仅具有高效性和准确性,还增强了实现的简便性,使该领域的非专业人员也能使用。
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引用次数: 0
Fast Numerical Solvers for Subdiffusion Problems with Spatial Interfaces 空间界面亚扩散问题的快速数值求解器
IF 1.1 4区 数学 Q2 Mathematics Pub Date : 2024-05-01 DOI: 10.4208/ijnam2024-1017
Boyang Yu,Yonghai Li, Jiangguo Liu
This paper develops novel fast numerical solvers for subdiffusion problems with spatial interfaces. These problems are modeled by partial differential equations that contain bothfractional order and conventional first order time derivatives. The former is non-local and approximated by L1 and L2 discretizations along with fast evaluation algorithms based on approximationby sums of exponentials. This results in an effective treatment of the “long-tail” kernel of subdiffusion. The latter is local and hence conventional implicit Euler or Crank-Nicolson discretizationscan be used. Finite volumes are utilized for spatial discretization based on consideration of localmass conservation. Interface conditions for mass and fractional fluxes are incorporated into thesefast solvers. Computational complexity and implementation procedures are briefly discussed.Numerical experiments demonstrate accuracy and efficiency of these new fast solvers.
本文针对具有空间界面的亚扩散问题开发了新型快速数值求解器。这些问题由包含分数阶和传统一阶时间导数的偏微分方程建模。前者是非局部的,通过 L1 和 L2 离散以及基于指数和近似的快速评估算法进行近似。这样就能有效处理亚扩散的 "长尾 "内核。后者是局部的,因此可以使用传统的隐式欧拉或 Crank-Nicolson 离散方法。在考虑局部质量守恒的基础上,利用有限体积进行空间离散。质量和分数通量的界面条件被纳入这些快速求解器。数值实验证明了这些新型快速求解器的精度和效率。
{"title":"Fast Numerical Solvers for Subdiffusion Problems with Spatial Interfaces","authors":"Boyang Yu,Yonghai Li, Jiangguo Liu","doi":"10.4208/ijnam2024-1017","DOIUrl":"https://doi.org/10.4208/ijnam2024-1017","url":null,"abstract":"This paper develops novel fast numerical solvers for subdiffusion problems with spatial interfaces. These problems are modeled by partial differential equations that contain both\u0000fractional order and conventional first order time derivatives. The former is non-local and approximated by L1 and L2 discretizations along with fast evaluation algorithms based on approximation\u0000by sums of exponentials. This results in an effective treatment of the “long-tail” kernel of subdiffusion. The latter is local and hence conventional implicit Euler or Crank-Nicolson discretizations\u0000can be used. Finite volumes are utilized for spatial discretization based on consideration of local\u0000mass conservation. Interface conditions for mass and fractional fluxes are incorporated into these\u0000fast solvers. Computational complexity and implementation procedures are briefly discussed.\u0000Numerical experiments demonstrate accuracy and efficiency of these new fast solvers.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
CFIRM: An Integrated Code Package for the Low-Temperature Plasma Simulation on Structured Grids CFIRM:结构网格低温等离子体模拟集成代码包
IF 1.1 4区 数学 Q2 Mathematics Pub Date : 2024-05-01 DOI: 10.4208/ijnam2024-1015
Jinwei Bai,Hongtao Liu,Xiaoming He,Wei Jiang, Yong Cao
This paper presents a recently developed full kinetic particle simulation code package, which is a two-dimensional highly integrated and universal framework for low-temperatureplasma simulation on both Cartesian and axisymmetric coordinate systems. This code packageis named CFIRM, since it is designed based on the continuous Galerkin immersed-finite-element(IFE) particle-in-cell (PIC) model with the polynomial-preserving-recovery (PPR) technique andthe Monte-Carlo-collision (MCC) method. Both the traditional and implicit PIC methods wereimplemented in the package. Incorporating the advantages of all these methods together, theCFIRM code can adopt explicit or implicit PIC schemes to track the motion trajectory of chargedparticles and deal with the collisions between plasma and neutral gas. Additionally, it can conveniently handle complex interface problems on structured grids. The CFRIM code has excellentversatility in low-temperature plasma simulation and can easily extend to various particle processing modules, such as the variable weights and adaptive particle management algorithms whichwere incorporated into this code to reduce the memory utilization rate. The implementation forthe main algorithms and the overall simulation framework of the CFIRM code package are rigorously described in details. Several simulations of the benchmark cases are carried out to validatethe reliability and accuracy of the CFIRM code. Moreover, two typical low-temperature plasmaengineering problems are simulated, including a hall thruster and a capacitively coupled plasmareactor, which demonstrates the applicability of this code package.
本文介绍了最近开发的全动力学粒子模拟代码包,它是一个二维高度集成的通用框架,可用于笛卡尔坐标系和轴对称坐标系上的低温等离子体模拟。该代码包被命名为 CFIRM,因为它是基于连续伽勒金沉浸-有限元(IFE)粒子-单元(PIC)模型、多项式保留-恢复(PPR)技术和蒙特-卡罗-碰撞(MCC)方法设计的。该软件包同时采用了传统和隐式 PIC 方法。结合所有这些方法的优点,CFIRM 代码可以采用显式或隐式 PIC 方案来跟踪带电粒子的运动轨迹,并处理等离子体与中性气体之间的碰撞。此外,它还能方便地处理结构网格上的复杂界面问题。CFRIM 代码在低温等离子体仿真中具有出色的通用性,并能轻松扩展到各种粒子处理模块,例如为降低内存利用率而加入的可变权重和自适应粒子管理算法。本文详细介绍了 CFIRM 代码包的主要算法和整体仿真框架。为了验证 CFIRM 代码的可靠性和准确性,对基准案例进行了多次仿真。此外,还模拟了两个典型的低温等离子体工程问题,包括霍尔推进器和电容耦合等离子体反应器,证明了该代码包的适用性。
{"title":"CFIRM: An Integrated Code Package for the Low-Temperature Plasma Simulation on Structured Grids","authors":"Jinwei Bai,Hongtao Liu,Xiaoming He,Wei Jiang, Yong Cao","doi":"10.4208/ijnam2024-1015","DOIUrl":"https://doi.org/10.4208/ijnam2024-1015","url":null,"abstract":"This paper presents a recently developed full kinetic particle simulation code package, which is a two-dimensional highly integrated and universal framework for low-temperature\u0000plasma simulation on both Cartesian and axisymmetric coordinate systems. This code package\u0000is named CFIRM, since it is designed based on the continuous Galerkin immersed-finite-element\u0000(IFE) particle-in-cell (PIC) model with the polynomial-preserving-recovery (PPR) technique and\u0000the Monte-Carlo-collision (MCC) method. Both the traditional and implicit PIC methods were\u0000implemented in the package. Incorporating the advantages of all these methods together, the\u0000CFIRM code can adopt explicit or implicit PIC schemes to track the motion trajectory of charged\u0000particles and deal with the collisions between plasma and neutral gas. Additionally, it can conveniently handle complex interface problems on structured grids. The CFRIM code has excellent\u0000versatility in low-temperature plasma simulation and can easily extend to various particle processing modules, such as the variable weights and adaptive particle management algorithms which\u0000were incorporated into this code to reduce the memory utilization rate. The implementation for\u0000the main algorithms and the overall simulation framework of the CFIRM code package are rigorously described in details. Several simulations of the benchmark cases are carried out to validate\u0000the reliability and accuracy of the CFIRM code. Moreover, two typical low-temperature plasma\u0000engineering problems are simulated, including a hall thruster and a capacitively coupled plasma\u0000reactor, which demonstrates the applicability of this code package.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Continuous/Discontinuous Finite Element Approximation of a 2d Navier-Stokes Problem Arising in Fluid Confinement 流体封闭中出现的二维 Navier-Stokes 问题的连续/非连续有限元逼近
IF 1.1 4区 数学 Q2 Mathematics Pub Date : 2024-05-01 DOI: 10.4208/ijnam2024-1013
Hermenegildo Borges De Oliveira, Nuno David Lopes
In this work, a stationary 2d Navier-Stokes problem with nonlinear feedback forcesfield is considered in the stream-function formulation. We use the Continuous/Discontinuous Finite Element Method (CD-FEM), with interior penalty terms, to numerically solve the associatedboundary-value problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stabilityand convergence of the method are also shown analytically. To validate the numerical modelregarding its applicability and robustness, several test cases are carried out.
在这项研究中,我们采用流函数公式考虑了一个具有非线性反馈力场的静态 2d Navier-Stokes 问题。我们使用带内部惩罚项的连续/非连续有限元法(CD-FEM)来数值求解相关的边界值问题。对于相关的连续和离散问题,我们证明了弱解的存在,并建立了它们唯一性的条件。我们还通过分析证明了该方法的一致性、稳定性和收敛性。为了验证数值模型的适用性和鲁棒性,我们进行了几个测试案例。
{"title":"Continuous/Discontinuous Finite Element Approximation of a 2d Navier-Stokes Problem Arising in Fluid Confinement","authors":"Hermenegildo Borges De Oliveira, Nuno David Lopes","doi":"10.4208/ijnam2024-1013","DOIUrl":"https://doi.org/10.4208/ijnam2024-1013","url":null,"abstract":"In this work, a stationary 2d Navier-Stokes problem with nonlinear feedback forces\u0000field is considered in the stream-function formulation. We use the Continuous/Discontinuous Finite Element Method (CD-FEM), with interior penalty terms, to numerically solve the associated\u0000boundary-value problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stability\u0000and convergence of the method are also shown analytically. To validate the numerical model\u0000regarding its applicability and robustness, several test cases are carried out.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
期刊
International Journal of Numerical Analysis and Modeling
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