Computers & Mathematics with Applications最新文献

英文中文

On full linear convergence and optimal complexity of adaptive FEM with inexact solver 非精确求解自适应有限元法的全线性收敛性和最优复杂度

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-27 DOI: 10.1016/j.camwa.2024.12.013

Philipp Bringmann, Michael Feischl, Ani Miraçi, Dirk Praetorius, Julian Streitberger

The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computation time. To this end, algorithmically, the standard adaptive finite element method (AFEM) integrates an inexact solver and nested iterations with discerning stopping criteria balancing the different error components. The analysis ensuring optimal convergence order of AFEM with respect to the overall computational cost critically hinges on the concept of R-linear convergence of a suitable quasi-error quantity. This work tackles several shortcomings of previous approaches by introducing a new proof strategy. Previously, the analysis of the algorithm required several parameters to be fine-tuned. This work leaves the classical reasoning and introduces a summability criterion for R-linear convergence to remove restrictions on those parameters. Second, the usual assumption of a (quasi-)Pythagorean identity is replaced by the generalized notion of quasi-orthogonality from Feischl (2022) [22]. Importantly, this paves the way towards extending the analysis of AFEM with inexact solver to general inf-sup stable problems beyond the energy minimization setting. Numerical experiments investigate the choice of the adaptivity parameters.

偏微分方程的任何数值格式的最终目标是在准最小的计算时间内计算出用户规定精度的近似值。为此，在算法上，标准的自适应有限元法（AFEM）集成了一个不精确求解器和嵌套迭代，并具有识别停止准则来平衡不同的误差分量。保证AFEM在总计算代价下的最优收敛顺序的分析，关键取决于合适准误差量的r -线性收敛概念。这项工作通过引入一种新的证明策略来解决以前方法的几个缺点。以前，对算法的分析需要对几个参数进行微调。本文在经典推理的基础上，引入了r -线性收敛的可和准则，消除了对这些参数的限制。其次，通常的（拟）毕达哥拉斯恒等式假设被Feischl(2022)[22]的广义拟正交概念所取代。重要的是，这为将非精确求解器的AFEM分析扩展到能量最小化设置之外的一般中频稳定问题铺平了道路。数值实验研究了自适应参数的选择。

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引用次数: 0

Unconditional superconvergence analysis of a novel energy dissipation nonconforming Crank-Nicolson FEM for Sobolev equations with high order Burgers' type nonlinearity 具有高阶Burgers型非线性Sobolev方程的一种新型能量耗散非协调Crank-Nicolson有限元法的无条件超收敛分析

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-20 DOI: 10.1016/j.camwa.2024.12.010

Tiantian Liang , Dongyang Shi

A novel energy dissipation Crank-Nicolson (C-N) fully discrete scheme is established by low order nonconforming

E Q_{1}^{r o t}

element for solving the Sobolev equations with high order Burgers' type nonlinearity. Firstly, the boundedness of the discrete solution in the broken

H^{1}

-norm is achieved directly by the energy dissipation property without using the known time-space splitting technique in the existing literatures, and its well-posedness is demonstrated by the Brouwer fixed point theorem. Secondly, by utilizing the special characters of nonconforming

E Q_{1}^{r o t}

element, the unconditional superclose result of order

O (h^{2} + τ^{2})

in the broken

H^{1}

-norm is gained strictly with no restrictions between the spatial partition parameter h and the time step τ. Moreover, the corresponding global superconvergent error estimate of order

O (h^{2} + τ^{2})

is proved by applying an interpolation post-processing approach. Thirdly, an application to some different finite elements and nonlinear PDEs is discussed, which shows that the proposed scheme and the analysis presented herein can be considered as a general framework to cope with. Lastly, the theoretical results are validated by four numerical examples.

针对具有高阶Burgers型非线性的Sobolev方程，利用低阶不协调EQ1rot元建立了一种新的能量耗散Crank-Nicolson （C-N）全离散格式。首先，不使用现有文献中已知的时空分裂技术，直接利用能量耗散特性获得了h1 -范数破碎离散解的有界性，并利用browwer不动点定理证明了其适定性；其次，利用非一致性EQ1rot元的特殊性质，在空间划分参数h与时间步长τ之间没有限制的情况下，严格地得到了破碎h1范数O（h2+τ2）阶的无条件超接近结果。此外，应用插值后处理方法证明了相应的O（h2+τ2）阶全局超收敛误差估计。第三，讨论了几种不同的有限元和非线性偏微分方程的应用，表明本文提出的方案和分析可以作为一个通用的框架来处理。最后，通过4个算例对理论结果进行了验证。

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引用次数: 0

Energy-preserving RERK-FEM for the regularized logarithmic Schrödinger equation 正则对数Schrödinger方程的能量守恒rk - fem

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-19 DOI: 10.1016/j.camwa.2024.12.009

Changhui Yao , Lei Li , Huijun Fan , Yanmin Zhao

A high-order implicit–explicit (IMEX) finite element method with energy conservation is constructed to solve the regularized logarithmic Schrödinger equation (RLogSE) with a periodic boundary condition. The discrete scheme consists of the relaxation-extrapolated Runge–Kutta (RERK) method in the temporal direction and the finite element method in the spatial direction. Choosing a proper relaxation parameter for the RERK method is the key technique for energy conservation. The optimal error estimates in the

L^{2}

-norm and

H^{1}

-norm are provided without any restrictions between time step size τ and mesh size h by temporal–spatial splitting technology. Numerical examples are given to demonstrate the theoretical results.

针对具有周期边界条件的正则对数Schrödinger方程（RLogSE），构造了一种具有能量守恒的高阶隐显有限元方法。离散格式包括时间方向上的松弛外推龙格-库塔（RERK）法和空间方向上的有限元法。选择合适的松弛参数是实现能量守恒的关键技术。利用时空分割技术，在不受时间步长τ和网格尺寸h限制的情况下，给出了l2范数和h1范数的最优误差估计。数值算例验证了理论结果。

引用次数: 0

A study of an efficient numerical method for solving the generalized fractional reaction-diffusion model involving a distributed-order operator along with stability analysis 涉及分布阶算子的广义分数阶反应扩散模型的有效数值求解方法研究及稳定性分析

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-18 DOI: 10.1016/j.camwa.2024.12.006

Muhammad Suliman , Muhammad Ibrahim , Ebrahem A. Algehyne , Vakkar Ali

In this manuscript, we study a generalized fractional reaction-diffusion model involving a distributed-order operator. An efficient hybrid approach is proposed to solve the presented model. The L1 approximation is utilized to discretize the time variable, while the mixed finite element method is employed for spatial discretization. A detailed error and stability analysis of the proposed method is provided. Furthermore, we prove that the computational accuracy achieved by the proposed approach is of order

O (h^{2} + {(Δ t)}^{3 - ξ_{\max}})

. To validate and evaluate the numerical approach, three numerical experiments are conducted, with results presented through graphs and tables.

在本文中，我们研究了一个涉及分布阶算子的广义分数反应扩散模型。提出了一种有效的混合方法来求解该模型。采用L1近似对时间变量进行离散，采用混合有限元法对空间变量进行离散。对该方法进行了详细的误差分析和稳定性分析。此外，我们证明了该方法的计算精度为O（h2+（Δt）3−ξmax）阶。为了验证和评价数值方法，进行了三个数值实验，并以图表的形式给出了结果。

引用次数: 0

Proving the stability estimates of variational least-squares kernel-based methods 证明了基于变分最小二乘核方法的稳定性估计

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-18 DOI: 10.1016/j.camwa.2024.12.008

Meng Chen , Leevan Ling , Dongfang Yun

Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.

出于对基于变分最小二乘内核的二阶椭圆偏微分方程求解方法的数值稳定性进行严格分析的需要，我们提供了以前缺乏的稳定性不等式。这填补了以前的工作[Comput. Math. Appl.有了现在严格证明的稳定性估计，我们就完成了理论基础，并将收敛行为与已证明的速率进行了比较。此外，我们还建立了另一个涉及加权离散规范的稳定性不等式，并提供了一个理论证明，证明基于加权最小二乘内核的配准方法收敛时并不需要精确的正交权重。我们新颖的理论见解得到了数值实例的验证，这些实例展示了这些方法在大网格比数据集上的相对效率和准确性。结果证实了我们对基于变分最小二乘法核的方法、基于最小二乘法核的配准方法以及我们新的基于加权最小二乘法核的配准方法性能的理论预测。最重要的是，我们的结果表明所有方法都以相同的速度收敛，验证了我们已证明的理论中的加权最小二乘法收敛理论。

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引用次数: 0

Strong approximation of the time-fractional Cahn–Hilliard equation driven by a fractionally integrated additive noise 分数积分加性噪声驱动的时间分数Cahn-Hilliard方程的强逼近

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-16 DOI: 10.1016/j.camwa.2024.12.007

Mariam Al-Maskari, Samir Karaa

In this article, we present a numerical scheme for solving a time-fractional stochastic Cahn–Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order

α \in (0, 1)

and a fractional time-integral noise of order

γ \in [0, 1]

. Our numerical approach combines a piecewise linear finite element method in space with a convolution quadrature in time, designed to handle both time-fractional operators, along with the

L^{2}

-projection for the noise. We conduct a detailed analysis of both spatially semidiscrete and fully discrete schemes, deriving strong convergence rates through energy-based arguments. The solution's temporal Hölder continuity played a key role in the error analysis. Unlike the stochastic Allen–Cahn equation, the inclusion of the unbounded elliptic operator in front of the cubic nonlinearity in our model added complexity and challenges to the error analysis. To address these challenges, we introduce novel techniques and refined error estimates. We conclude with numerical examples that validate our theoretical findings.

在本文中，我们提出了一种数值方案，用于求解由加性分数积分高斯噪声驱动的时间分数随机卡恩-希利亚德方程。该模型涉及阶数为α∈(0,1)的卡普托分数时间导数和阶数为γ∈[0,1]的分数时间积分噪声。我们的数值方法结合了空间的分片线性有限元法和时间的卷积正交法，旨在处理这两个时间分数算子以及噪声的 L2 投影。我们对空间半离散和完全离散方案进行了详细分析，通过基于能量的论证得出了强大的收敛率。解的时间荷尔德连续性在误差分析中起到了关键作用。与随机 Allen-Cahn 方程不同，我们的模型在立方非线性前加入了无界椭圆算子，这增加了误差分析的复杂性和挑战性。为了应对这些挑战，我们引入了新技术和完善的误差估计。最后，我们通过数值示例验证了我们的理论发现。

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引用次数: 0

A high-precision numerical method based on spectral deferred correction for solving the time-fractional Allen-Cahn equation 基于谱延迟校正的高精度时间分数阶Allen-Cahn方程求解方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-12 DOI: 10.1016/j.camwa.2024.11.034

Jing Wang, Xuejuan Chen, Jinghua Chen

This paper presents a high-precision numerical method based on spectral deferred correction (SDC) for solving the time-fractional Allen-Cahn equation. In the temporal direction, we establish a stabilized variable-step L1 semi-implicit scheme which satisfies the discrete variational energy dissipation law and the maximum principle. Through theoretical analysis, we prove that this numerical scheme is convergent and unconditionally stable. In the spatial direction, we apply the Fourier-Galerkin spectral method for discretization and conduct an error analysis of the fully discretized scheme. Since the stabilized variable-step L1 semi-implicit scheme is only of first-order accuracy in the time direction, to improve the accuracy, we combine explicit and implicit schemes (linear terms are handled implicitly, while nonlinear terms are handled explicitly) to establish a stabilized semi-implicit spectral deferred correction scheme. Finally, we verify the validity and feasibility of the numerical scheme through numerical examples.

提出了一种基于谱延迟校正（SDC）的高精度数值求解时间分数阶Allen-Cahn方程的方法。在时间方向上，我们建立了满足离散变分能量耗散规律和极大值原则的稳定变步长L1半隐式格式。通过理论分析，证明了该数值格式是收敛且无条件稳定的。在空间方向上，我们采用傅里叶-伽辽金谱法进行离散化，并对完全离散化方案进行误差分析。由于稳定的变步长L1半隐式格式在时间方向上仅具有一阶精度，为了提高精度，我们将显式和隐式格式（隐式处理线性项，显式处理非线性项）结合起来，建立了稳定的半隐式频谱延迟校正格式。最后，通过数值算例验证了数值格式的有效性和可行性。

引用次数: 0

Meshfree methods for nonlinear equilibrium radiation diffusion equation with interface and discontinuous coefficient 具有界面和不连续系数的非线性平衡辐射扩散方程的无网格法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-03 DOI: 10.1016/j.camwa.2024.11.029

Haowei Liu, Zhiyong Liu, Qiuyan Xu, Jiye Yang

The partial differential equation describing equilibrium radiation diffusion is strongly nonlinear, which has been widely utilized in various fields such as astrophysics and others. The equilibrium radiation diffusion equation usually appears over multiple complicated domains, and the material characteristics vary between each domain. The diffusion coefficient near the interface is discontinuous. In this paper, the equilibrium radiation diffusion equation with discontinuous diffusion coefficient will be solved numerically by the unsymmetric radial basis function collocation method. The energy term

T^{4}

is linearized by utilizing the Picard-Newton and Richtmyer linearization methods on the basis of the fully implicit scheme discretization. And the successive permutation iteration and direct linearization methods are applied to linearize the diffusion terms. The accuracy of the proposed methods is proved by numerical experiments for regular and irregular domains with different types of interfaces.

描述平衡辐射扩散的偏微分方程是一种强非线性方程，已广泛应用于天体物理等各个领域。平衡辐射扩散方程通常出现在多个复杂的区域上，而材料的特性在每个区域之间是不同的。界面附近的扩散系数不连续。本文采用非对称径向基函数配点法对具有不连续扩散系数的平衡辐射扩散方程进行数值求解。在全隐式格式离散化的基础上，利用Picard-Newton和richmyer线性化方法对能量项T4进行线性化。采用逐次置换迭代法和直接线性化法对扩散项进行线性化处理。数值实验证明了该方法在具有不同类型界面的规则域和不规则域上的准确性。

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引用次数: 0

A graded mesh technique for numerical approximation of a multi-term Caputo time-fractional Fokker-Planck equation in 2D space 二维空间中多项Caputo时间分数型Fokker-Planck方程数值逼近的梯度网格技术

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-12-02 DOI: 10.1016/j.camwa.2024.11.031

Pradip Roul, Trishna Kumari, Sameer N. Khandagale

This paper focuses on the design of an efficient numerical approach for solving a two-dimensional multi-term Caputo time fractional Fokker-Planck (TFFP) model. The solution of such problem, in general, shows a weak singularity at the time origin. A numerical technique based on a graded time mesh is proposed to handle the singular behavior of the solution. The multi-term Caputo time fractional derivatives in the TFFP model are discretized by means of the L1 scheme on the nonuniform mesh, while a high-order compact alternating direction implicit finite difference scheme is designed to approximate the spatial derivatives. Convergence and stability analysis of the suggested method is analyzed. Two numerical examples subjected to smooth and nonsmooth exact solutions are presented to demonstrate the applicability and accuracy of the method. The results obtained by the proposed graded mesh technique are compared with the results obtained by the uniform mesh technique.

本文着重设计了一种求解二维多项卡普托时间分数阶Fokker-Planck （TFFP）模型的有效数值方法。一般情况下，这类问题的解在时间原点处表现出弱奇异性。提出了一种基于分级时间网格的数值方法来处理解的奇异性。在非均匀网格上采用L1格式对TFFP模型中的多项Caputo时间分数阶导数进行离散化，设计了一种高阶紧凑交替方向隐式有限差分格式来逼近空间导数。对该方法的收敛性和稳定性进行了分析。给出了光滑精确解和非光滑精确解的两个数值算例，验证了该方法的适用性和准确性。将梯度网格技术所得结果与均匀网格技术所得结果进行了比较。

引用次数: 0

A semi-analytic collocation technique for solving 3D anomalous non-linear thermal conduction problem associated with the Caputo fractional derivative 求解三维Caputo分数阶导数异常非线性热传导问题的半解析配置技术

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-11-30 DOI: 10.1016/j.camwa.2024.11.032

Farzaneh Safari , Yanjun Duan

A semi-analytic numerical method is described as an efficient meshless approach for the solution of anomalous non-linear thermal conduction problems in functionally graded materials in which the model results in fractional boundary value problems. The first key feature in this scheme is the derivation and discretization of the fractional derivative at every time step. The second key feature is the trigonometric basis functions (TBFs) as the basis functions were introduced by the need for approximate solutions on boundary conditions with more flexibility in choosing collocation points. Moreover, the approximate solution of the anomalous thermal conduction problems converges to the exact solution as γ is closed to 1 in the full closed time interval for three simulated numerical results.

半解析数值方法是解决功能梯度材料中异常非线性热传导问题的一种有效的无网格方法，其中该模型导致分数边值问题。该方案的第一个关键特征是在每个时间步长对分数阶导数进行求导和离散化。第二个关键特征是三角基函数（tbf），因为基函数是由边界条件近似解的需要引入的，在选择搭配点时具有更大的灵活性。此外，对于三个模拟数值结果，当γ在全封闭时间区间内趋近于1时，异常热传导问题的近似解收敛于精确解。

引用次数: 0

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