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Composite iteration for isogeometric collocation method using LSPIA and Schulz iteration 基于LSPIA和Schulz迭代的等几何配点法复合迭代

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-01-09 DOI: 10.1016/j.camwa.2024.12.026

Gengchen Li, Hongwei Lin, Depeng Gao

The isogeometric least-squares collocation method (IGA-L) is an effective numerical technique for solving partial differential equations (PDEs), which utilizes the non-uniform rational B-splines (NURBS) to represent the numerical solution and constructs a system of equations using more collocation points than the number of unknowns. However, on the one hand, the convergence rate of the isogeometric collocation method is lower than that of the isogeometric Galerkin (IGA-G) method; on the other hand, the freedom of the numerical solution cannot be determined in advance to reach specified precision. In this paper, we model the solution of PDEs using IGA-L as a data fitting problem, in which the linear combination of the numerical solution and its derivatives is employed to fit the load function. Moreover, we develop a composite iterative method combining the least-squares progressive-iterative approximation (LSPIA) with the three-order Schulz iteration to solve the data fitting problem. The convergence of composite iterative method is proved, and the error bound is analyzed. Numerical results demonstrate that the convergence rate of the composite iterative method for IGA-L is nearly the same as that of IGA-G. Finally, we propose an incremental fitting algorithm with the composite iterative method, by which the freedom of numerical solution can be determined automatically to reach the specified fitting precision.

等距最小二乘配位法（IGA-L）是求解偏微分方程（PDE）的一种有效数值技术，它利用非均匀有理 B-样条曲线（NURBS）来表示数值解，并用多于未知数个数的配位点来构造方程组。然而，一方面，等距配位法的收敛速度低于等距 Galerkin（IGA-G）法；另一方面，数值解的自由度无法事先确定，无法达到指定精度。在本文中，我们将使用 IGA-L 求解 PDEs 作为一个数据拟合问题进行建模，利用数值解及其导数的线性组合来拟合载荷函数。此外，我们还开发了一种将最小二乘渐进迭代近似（LSPIA）与三阶舒尔茨迭代相结合的复合迭代法来解决数据拟合问题。证明了复合迭代法的收敛性，并分析了误差约束。数值结果表明，IGA-L 复合迭代法的收敛速率与 IGA-G 几乎相同。最后，我们提出了一种采用复合迭代法的增量拟合算法，通过该算法可以自动确定数值解的自由度，从而达到指定的拟合精度。

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引用次数: 0

Modified preconditioned Newton-Krylov approaches for Navier-Stokes equations using nodal integral method 用节点积分法求解Navier-Stokes方程的修正预条件Newton-Krylov逼近

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-01-09 DOI: 10.1016/j.camwa.2024.12.027

Nadeem Ahmed , Suneet Singh , Ram Prakash Bharti

Nodal integral methods (NIMs) have been proven effective in solving a wide range of scientific and engineering problems by providing accurate solutions with coarser grids. Despite notable advantages, these methods have encountered limited acceptance within the fluid flow community, primarily due to the lack of robust and efficient nonlinear solvers for the algebraic equations arising from discretization using NIM. A preconditioned Jacobian-free Newton-Krylov approach has been recently developed to solve Navier-Stokes equations to overcome this limitation. The developed approach has extended the acceptability of NIM and demonstrated considerable gains in computational time. However, a challenge persists in the efficiency of the proposed approach, particularly in solving the pressure Poisson equation. Addressing this, we offer novel strategies and algorithms to solve the pressure Poisson equation. These strategies aim to improve the computational efficiency of NIMs, making them more effective in solving complex problems in scientific and engineering applications.

节点积分方法（nim）通过在较粗糙的网格上提供精确的解，在解决广泛的科学和工程问题方面已被证明是有效的。尽管具有显著的优势，但这些方法在流体学界的接受程度有限，主要是由于使用NIM进行离散化产生的代数方程缺乏鲁棒和有效的非线性求解器。为了克服这一限制，最近开发了一种无雅可比预条件牛顿-克雷洛夫方法来求解Navier-Stokes方程。所开发的方法扩展了NIM的可接受性，并在计算时间上取得了可观的收益。然而，所提出的方法的效率仍然存在挑战，特别是在求解压力泊松方程时。为了解决这个问题，我们提供了新的策略和算法来求解压力泊松方程。这些策略旨在提高NIMs的计算效率，使其更有效地解决科学和工程应用中的复杂问题。

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引用次数: 0

MC-CDNNs: The Monte Carlo-coupled deep neural networks approach for stochastic dual-porosity-Stokes flow coupled model MC-CDNNs：随机双孔隙度- stokes流耦合模型的Monte carlo耦合深度神经网络方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-01-08 DOI: 10.1016/j.camwa.2024.12.024

Jian Li , Shaoxuan Li , Jing Yue

In this paper, we propose a coupled neural network learning method to solve the stochastic dual-porosity-Stokes flow problem. We combine Monte Carlo and coupled deep neural networks methods (MC-CDNNs) to transform the uncertain stochastic coupled problems into a deterministic coupled problem, and compile the complex interface conditions associated with the coupled problem into the neural network to guarantee the physical constraints of the approximate solution. In addition, the convergence analysis illustrates the capability of the method in solving the stochastic coupling problem. Particularly, we conducted 2D/3D numerical experiments to demonstrate the algorithm's effectiveness and efficiency, and to show its advantages in practical applications.

本文提出了一种耦合神经网络学习方法来求解随机双孔隙度-斯托克斯流问题。将蒙特卡罗方法与耦合深度神经网络方法（MC-CDNNs）相结合，将不确定随机耦合问题转化为确定性耦合问题，并将与耦合问题相关的复杂界面条件编译到神经网络中，以保证近似解的物理约束。此外，收敛性分析说明了该方法在求解随机耦合问题中的能力。通过二维/三维数值实验验证了该算法的有效性和高效性，并在实际应用中展示了其优势。

引用次数: 0

Understanding avascular tumor growth and drug interactions through numerical analysis: A finite element method approach 通过数值分析了解无血管肿瘤生长和药物相互作用：一种有限元方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-01-08 DOI: 10.1016/j.camwa.2024.12.023

Vivek S. Yadav , Nishant Ranwan , Nagaiah Chamakuri

This article establishes the existence of a fully discrete weak solution for the tumor growth model, which is described by a coupled non-linear reaction-diffusion system. This model incorporates crucial elements such as cellular proliferation, nutrient diffusion, prostate-specific antigen, and drug effects. We employ the finite element method for spatial discretization and the implicit Euler method for temporal discretization. Firstly, we analyzed the existence and uniqueness of the fully discretized tumor growth model. Additionally, stability bounds for the fully discrete coupled system are derived. Secondly, through multiple numerical simulations utilizing higher-order finite element methods, we analyze tumor growth behavior both with and without drug interaction, yielding a more accurate numerical solution. Furthermore, we compare CPU time efficiency across different time marching methods and explore various preconditioners to optimize computational performance.

本文确定了肿瘤生长模型存在一个完全离散的弱解，该模型由一个耦合非线性反应-扩散系统描述。该模型包含了细胞增殖、营养扩散、前列腺特异性抗原和药物效应等关键要素。我们采用有限元法进行空间离散化，隐式欧拉法进行时间离散化。首先，我们分析了完全离散化肿瘤生长模型的存在性和唯一性。此外，还推导出了完全离散耦合系统的稳定性边界。其次，我们利用高阶有限元方法进行了多次数值模拟，分析了有药物相互作用和无药物相互作用的肿瘤生长行为，得出了更精确的数值解。此外，我们还比较了不同时间行进方法的 CPU 时间效率，并探索了各种预处理方法，以优化计算性能。

引用次数: 0

Numerical simulation of nonlinear fractional integro-differential equations on two-dimensional regular and irregular domains: RBF partition of unity 二维正则和不规则区域上非线性分数阶积分微分方程的数值模拟：单位的RBF划分

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-01-08 DOI: 10.1016/j.camwa.2024.12.019

M. Fardi , B. Azarnavid , S. Mohammadi

In this article, we introduce a numerical method that combines local radial basis functions partition of unity with backward differentiation formula to efficiently solve linear and nonlinear fractional integro-differential equations on two-dimensional regular and irregular domains. We derive the time-discretized formulation using the backward difference formula. The meshless radial basis function method, particularly the radial basis function partition of unity method, offers advantages such as flexibility, accuracy, ease of implementation, adaptive refinement, and efficient parallelization. We apply the radial basis function partition of unity method to spatially discretize the problem using the scaled Lagrangian form of polyharmonic splines as approximation bases. Numerical simulations demonstrate the efficacy of our method in solving linear and nonlinear fractional integro-differential equations with complex domains and smooth and nonsmooth initial conditions. Comparative analysis confirms the superior performance of our proposed method.

本文介绍了一种将局部径向基函数的单位划分与后向微分公式相结合的数值方法，有效地求解二维正则和不规则域上的线性和非线性分数阶积分微分方程。我们利用后向差分公式推导出时间离散公式。无网格径向基函数法，特别是单元法的径向基函数划分，具有灵活、准确、易于实现、自适应细化和高效并行化等优点。以多谐样条的缩放拉格朗日形式为近似基，采用统一的径向基函数划分方法对问题进行空间离散化。数值模拟证明了该方法在求解具有复杂域、光滑和非光滑初始条件的线性和非线性分数阶积分微分方程方面的有效性。对比分析证实了该方法的优越性。

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引用次数: 0

Superconvergence of triquadratic finite elements for the second-order elliptic equation with variable coefficients 二阶变系数椭圆型方程三二次元的超收敛性

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2025-01-08 DOI: 10.1016/j.camwa.2025.01.001

Jinghong Liu

This study focuses on superconvergence of the tensor-product quadratic finite element (so-called triquadratic finite element) in a regular family of rectangular partitions of the domain for the second-order elliptic equation with variable coefficients in three dimensions. In this paper, we first introduce a variable coefficients elliptic boundary value problem and its finite elements discretization, as well as some important functions such as the discrete Green's function and discrete derivative Green's function. Then, an interpolation operator of project type is given, by which we derive a interpolation fundamental estimate (so-called weak estimate) for general variable coefficients elliptic equations. Finally, combining the weak estimate and estimates for the discrete Green's function and discrete derivative Green's function, we get superconvergence estimates for derivatives and function values of the finite element approximation in the pointwise sense of the

L^{\infty}

-norm.

本文研究了三维变系数二阶椭圆方程的张量积二次元（所谓的三二次元）在正则矩形分区域上的超收敛性。本文首先介绍了一类变系数椭圆型边值问题及其有限元离散化，以及离散格林函数和离散导数格林函数等重要函数。然后，给出了一个项目型插值算子，并由此导出了一般变系数椭圆方程的插值基本估计（即弱估计）。最后，结合离散格林函数和离散导数格林函数的弱估计和估计，得到了L∞范数点向意义上有限元逼近的导数和函数值的超收敛估计。

引用次数: 0

Numerical wetting simulations using the plicRDF-isoAdvector unstructured Volume-of-Fluid (VOF) method 使用plicrdf -等矢量非结构流体体积（VOF）方法的数值润湿模拟

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-31 DOI: 10.1016/j.camwa.2024.12.015

Muhammad Hassan Asghar, Mathis Fricke, Dieter Bothe, Tomislav Marić

Numerical simulation of wetting and dewetting of geometrically complex surfaces benefits from the boundary-fitted unstructured Finite Volume method because it discretizes boundary conditions on geometrically complex domain boundaries with second-order accuracy and simplifies the simulation workflow. The plicRDF-isoAdvector method, an unstructured geometric Volume-of-Fluid (VOF) method, reconstructs the Piecewise Linear Interface Calculation (PLIC) interface by reconstructing signed distance functions (RDF). This method is chosen to investigate wetting processes because of its volume conservation property and high computational efficiency. The present work verifies and validates the plicRDF-isoAdvector method for wetting problems, employing five different case studies. The first study investigates the accuracy of the interface advection near walls. The method is further investigated for the spreading of droplets on a flat and a spherical surface, respectively, for which excellent agreement with the reference solutions is obtained. Furthermore, a validation study using a droplet spreading test case is carried out. The uncompensated Young stress is introduced in the contact angle boundary condition, which significantly improves the validation of the numerical method. Furthermore, a 2D capillary rise is considered, and a numerical comparison based on results from previous work is performed. A suite with all case studies, input data, and Jupyter Notebooks used in this study are publicly available to facilitate further research and comparison with other numerical codes.

拟合边界的非结构有限体积法以二阶精度离散了几何复杂曲面的边界条件，简化了模拟工作流程，有利于几何复杂曲面的润湿和脱湿数值模拟。plicRDF-isoAdvector方法是一种非结构化几何流体体积（VOF）方法，通过重构符号距离函数（RDF）来重构分段线性界面计算（PLIC）界面。由于该方法具有体积守恒性和计算效率高的特点，因此选择该方法来研究润湿过程。目前的工作验证了plicRDF-isoAdvector方法在润湿问题上的有效性，采用了五个不同的案例研究。第一项研究考察了壁面界面平流的准确性。进一步研究了液滴在平面和球面上的扩散，得到了与参考溶液非常吻合的结果。此外，利用液滴扩散试验案例进行了验证研究。在接触角边界条件中引入无补偿杨氏应力，大大提高了数值方法的有效性。此外，考虑了二维毛细上升，并根据以往工作的结果进行了数值比较。本研究中使用的所有案例研究、输入数据和Jupyter notebook的套件都是公开的，以促进进一步的研究和与其他数字代码的比较。

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引用次数: 0

Robust finite element solvers for distributed hyperbolic optimal control problems 分布式双曲型最优控制问题的鲁棒有限元求解方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-30 DOI: 10.1016/j.camwa.2024.12.012

Ulrich Langer , Richard Löscher , Olaf Steinbach , Huidong Yang

We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic distributed, tracking-type optimal control problems with both the standard

L^{2}

and the more general energy regularizations. In contrast to the usual time-stepping approach, we discretize the optimality system by space-time continuous piecewise-linear finite element basis functions which are defined on fully unstructured simplicial meshes. If we aim at the asymptotically best approximation of the given desired state

y_{d}

by the computed finite element state

y_{ϱ h}

, then the optimal choice of the regularization parameter ϱ is linked to the space-time finite element mesh-size h by the relations

ϱ = h^{4}

and

ϱ = h^{2}

for the

L^{2}

and the energy regularization, respectively. For this setting, we can construct robust (parallel) iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in the case of adaptive mesh refinements. The numerical results illustrate the theoretical findings firmly.

我们提出，分析，并测试了新的鲁棒迭代求解器，用于线性代数方程组，这些方程组由简化最优性系统的时空有限元离散产生，定义了双曲分布的近似解，跟踪型最优控制问题具有标准L2和更一般的能量正则化。与通常的时间步进方法不同，我们采用在完全非结构简单网格上定义的时空连续分段线性有限元基函数来离散最优性系统。如果我们的目标是通过计算的有限元状态yϱh逼近给定的期望状态yd，那么正则化参数ϱ的最优选择将分别通过L2和能量正则化的ϱ=h4和ϱ=h2关系与时空有限元网格尺寸h联系起来。在这种情况下，我们可以为简化的有限元最优性系统构建鲁棒（并行）迭代求解器。这些结果可以推广到适应网格尺寸局部行为的可变正则化参数，这些参数在自适应网格细化的情况下可能会发生很大变化。数值结果有力地证明了理论结论。

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引用次数: 0

Vectorized implementation of primal hybrid FEM in MATLAB 原始混合有限元的MATLAB矢量化实现

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-30 DOI: 10.1016/j.camwa.2024.12.017

Harish Nagula Mallesham , Kamana Porwal , Jan Valdman , Sanjib Kumar Acharya

We present efficient MATLAB implementations of the lowest-order primal hybrid finite element method (FEM) for linear second-order elliptic and parabolic problems with mixed boundary conditions in two spatial dimensions. We employ backward Euler and the Crank-Nicolson finite difference scheme for the complete discrete setup of the parabolic problem. All the codes presented are fully vectorized using matrix-wise array operations. Numerical experiments are conducted to show the performance of the software.

本文给出了二维空间中具有混合边界条件的线性二阶椭圆型和抛物型问题的最低阶原始混合有限元方法的MATLAB实现。对于抛物型问题的完全离散解，我们采用了后向欧拉格式和Crank-Nicolson有限差分格式。所有的代码都是完全矢量化的，使用矩阵数组操作。通过数值实验验证了该软件的性能。

引用次数: 0

Semi-analytical algorithm for quasicrystal patterns 准晶图的半解析算法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-30 DOI: 10.1016/j.camwa.2024.12.016

Keyue Sun, Xiangjie Kong, Junxiang Yang

To efficiently simulate the quasicrystal patterns, we present a multi-stage semi-analytically algorithm. Utilizing the operator splitting strategy, we first split the original equation into three subproblems. A second-order five-stage scheme consists of solving four nonlinear ordinary differential equations with half time step and solving a linear partial differential equation with full time step. Using the methods of separation of variables, the nonlinear ODEs have analytical solutions. The linear PDE can also be analytically solved by using the Fourier-spectral method in space. In this sense, our proposed is semi-analytical because we only adopt an approximation in time. In each time step, we only need to compute several analytically solutions in a step-by-step manner. Therefore, the algorithm will be highly efficient and the simulation can be easily implemented. The performance and high efficiency of our proposed algorithm are verified via several simulations. To facilitate the interested readers to develop related researches, a MATLAB code for generating 12-fold quasicrystal patterns is provided in Appendix. We also share the computational code on Code Ocean platform, please refer to https://doi.org/10.24433/CO.6028082.v1.

为了有效地模拟准晶图案，我们提出了一种多阶段半解析算法。利用算子拆分策略，首先将原方程拆分为三个子问题。二阶五阶段格式包括求解四个半时间步长非线性常微分方程和求解一个全时间步长线性偏微分方程。利用分离变量的方法，得到了非线性微分方程的解析解。线性偏微分方程也可以用空间傅里叶谱法解析求解。从这个意义上说，我们的建议是半解析的，因为我们只采用时间上的近似。在每个时间步中，我们只需要一步一步地计算几个解析解。因此，该算法效率高，且易于实现仿真。通过仿真验证了该算法的性能和高效性。为了方便有兴趣的读者开展相关研究，附录中提供了生成12重准晶图的MATLAB代码。我们也在code Ocean平台上分享计算代码，请参考https://doi.org/10.24433/CO.6028082.v1。

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