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Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon 多边形中的半线性椭圆偏微分方程的解析正则性和解法近似
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2024-01-11 DOI: 10.1007/s10092-023-00562-0
Yanchen He, Christoph Schwab

In an open, bounded Lipschitz polygon (Omega subset mathbb {R}^2), we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in (Omega ). The boundary conditions on each edge of (partial Omega ) are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in (H^1(Omega )), quantized tensor formats and certain deep neural networks.

在一个开放的、有界的 Lipschitz 多边形(Omega subset mathbb {R}^2)中,我们为一个半线性的、具有解析非线性的椭圆 PDE 建立了加权解析正则性,该 PDE 受制于一个在 (Omega ) 中解析的源项 f。(partial Omega )每条边上的边界条件要么是均相 Dirichlet,要么是均相 Neumann BC。目前确定的解的加权解析正则性意味着各种近似方案的指数收敛性:hp-有限元、通过 (H^1(Omega )) 中解集的 Kolmogorov n 宽的降阶模型、量化张量格式和某些深度神经网络。
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引用次数: 0
The Hermite-type virtual element method with interior penalty for the fourth-order elliptic problem 四阶椭圆问题的带内部惩罚的赫米特型虚拟元素法
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2024-01-03 DOI: 10.1007/s10092-023-00555-z
Jikun Zhao, Teng Chen, Bei Zhang, Xiaojing Dong

We present a Hermite-type virtual element method with interior penalty to solve the fourth-order elliptic problem over general polygonal meshes, where some interior penalty terms are added to impose the (C^1) continuity. A (C^0)-continuous Hermite-type virtual element with local (H^2) regularity is constructed, such that it can be used in the interior penalty scheme. We prove the boundedness of basis functions and interpolation error estimates of Hermite-type virtual element. After introducing a discrete energy norm, we present the optimal convergence of the interior penalty scheme. Compared with some existing methods, the proposed interior penalty method uses fewer degrees of freedom. Finally, we verify the theoretical results through some numerical examples.

我们提出了一种带有内部惩罚的 Hermite 型虚元方法,用于求解一般多边形网格上的四阶椭圆问题,其中添加了一些内部惩罚项来施加 (C^1) 连续性。我们构造了一个具有局部正则性的(H^2)连续赫米特型虚元,使其可以用于内部惩罚方案。我们证明了赫尔墨特型虚元的基函数有界性和插值误差估计。在引入离散能量规范后,我们提出了内部惩罚方案的最优收敛性。与现有的一些方法相比,所提出的内部惩罚方法使用了更少的自由度。最后,我们通过一些数值实例验证了理论结果。
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引用次数: 0
Error estimates of invariant-preserving difference schemes for the rotation-two-component Camassa–Holm system with small energy 小能量旋转两分量卡马萨-霍尔姆系统的保不变差分方案的误差估计
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2024-01-02 DOI: 10.1007/s10092-023-00558-w
Qifeng Zhang, Jiyuan Zhang, Zhimin Zhang

A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are (O(tau ^2 + h^2)) for the velocity in the (L^infty )-norm and the surface elevation in the (L^2)-norm, where (tau ) denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values.

最近提出了一个旋转-两分量卡马萨-霍尔姆(R2CH)系统,用于描述浅水波在重力影响下的运动。这是一个高度非线性和强耦合的偏微分方程系统。设计数值方案的一个关键问题是在离散层面尽可能多地保留不变式。在本文中,我们提出了一种可证明的隐式非线性差分方案,它至少保留了三个离散守恒不变式:能量、质量和动量,并通过布劳德定理证明了差分解的存在性。误差分析基于差分方案中对双线性算子的新颖而精细的估计。通过巧妙地使用能量法,我们证明了差分方案不仅在旋转参数减小时无条件收敛,而且在旋转参数不为零的小能量情况下收敛时没有任何步长比限制。对于 (L^infty )-正态的速度和 (L^2)-正态的表面高程,两种设置(旋转参数为零或非零)下的收敛阶数都是(O(tau ^2 + h^2)),其中 (tau )分别表示时间步长和 h 表示空间步长。适当设计的两级迭代方案证实了理论预测。与现有文献中的数值方法相比,所提出的方法证明了其在较大域上进行长时间模拟的有效性,以及对光滑和非光滑初始值的卓越分辨率。
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引用次数: 0
An algorithm for the spectral radius of weakly essentially irreducible nonnegative tensors 弱本质不可还原非负张量谱半径的算法
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2024-01-02 DOI: 10.1007/s10092-023-00561-1

Abstract

In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples.

摘要 本文首先定义了弱本质不可还原非负张量,并统一了本质正张量、弱正张量和广义弱正张量的定义。然后给出了一种基于隐式平移变换求弱本质不可还原非负张量谱半径的算法,并利用矩阵的有向图分析了算法的线性收敛条件,最后利用数值实例比较了相关算法的计算效率。
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引用次数: 0
$$L^2(I;H^1(Omega )^d)$$ and $$L^2(I;L^2(Omega )^d)$$ best approximation type error estimates for Galerkin solutions of transient Stokes problems 瞬态斯托克斯问题 Galerkin 解决方案的 $$L^2(I;H^1(Omega )^d)$$ 和 $$L^2(I;L^2(Omega )^d)$$ 最佳近似型误差估计值
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2023-12-29 DOI: 10.1007/s10092-023-00560-2
Dmitriy Leykekhman, Boris Vexler

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in (L^2(I;L^2(Omega )^d)) and (L^2(I;H^1(Omega )^d)) norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type (L^2(I;H^1(Omega ))) error estimate are new even for scalar parabolic problems.

本文为 (L^2(I;L^2(Omega )^d)) 和 (L^2(I;H^1(Omega )^d)) 规范下的瞬态斯托克斯问题全离散 Galerkin 解建立了最佳近似型估计。这些估计值填补了瞬态斯托克斯问题误差分析的空白,并有大量应用。分析自然扩展到非均质抛物面问题。最佳类型 (L^2(I;H^1(Omega ))) 误差估计即使对于标量抛物问题也是全新的。
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引用次数: 0
A convection–diffusion problem with a large shift on Durán meshes 杜兰网格上的大位移对流扩散问题
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2023-12-28 DOI: 10.1007/s10092-023-00559-9

Abstract

A convection–diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Durán type meshes, as well as on coarser Durán type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments.

摘要 研究了一个空间偏移较大的对流扩散问题。提供了在层适应杜兰型网格上以及在出现弱层的地方在更粗糙的杜兰型网格上对高阶有限元方法的数值分析。数值实验证实了理论结果。
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引用次数: 0
On the positivity of the discrete Green’s function for unstructured finite element discretizations in three dimensions 关于三维非结构化有限元离散的离散格林函数的实在性
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2023-12-22 DOI: 10.1007/s10092-023-00556-y
Andrew P. Miller

The aim of this paper is twofold. First, we prove (L^p) estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter h. Actually, we show that at the singularity the discrete Green’s function is of order (h^{-1}), which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.

本文有两个目的。首先,我们证明了三维正则化格林函数的 (L^p) 估计值。然后,我们建立了离散格林函数的新估计,并得到了一些正验结果。事实上,我们证明了在奇点处离散格林函数的阶(h^{-1}),这与连续格林函数的行为一致。此外,我们还证明离散格林函数是正的,并且在远离奇点处呈指数衰减。我们还提供了离散格林函数在 Delaunay 网格上的数值持续负值,这意味着离散哈纳克不等式无法在非结构化有限元离散中成立。
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引用次数: 0
A class of two stage multistep methods in solutions of time dependent parabolic PDEs 求解随时间变化的抛物型 PDE 的一类两阶段多步骤方法
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2023-12-21 DOI: 10.1007/s10092-023-00557-x
Moosa Ebadi, Mohammad Shahriari

In this manuscript, a new class of high-order multistep methods on the basis of hybrid backward differentiation formulas (BDF) have been illustrated for the numerical solutions of systems of ordinary differential equations (ODEs) arising from semi-discretization of time dependent partial differential equations. Order and stability analysis of the methods have been discussed in detail. By using an off-step point together with a step point in the first derivative of the solution, the new methods obtained are A-stable for order p, ((p=4,5,6,7)) and (A(alpha ))-stable for order p, ((p=8,9,ldots , 14).) Compared to the existing BDF based method, i.e. class (2+1,) hybrid BDF methods (HBDF), super-future points based methods (SFPBM) and MEBDF, there is a good improvement regarding to absolute stability regions and orders. Some numerical examples are given in order to check the advantage of these methods in reducing the CPU time and thus in increasing accuracy of low and high order the new methods compared to those of SFPBM and MEBDF.

本手稿以混合后向微分公式(BDF)为基础,阐述了一类新的高阶多步方法,用于数值求解由时间相关偏微分方程半离散化产生的常微分方程(ODE)系统。详细讨论了这些方法的阶次和稳定性分析。通过在解的一阶导数中使用离阶点和阶点,得到的新方法在阶数为p时是A稳定的((p=4,5,6,7)),在阶数为p时是(A(α))稳定的(((p=8,9,ldots , 14).)。与现有的基于BDF的方法(即类(2+1,)混合BDF方法(HBDF)、基于超未来点的方法(SFPBM)和MEBDF)相比,在绝对稳定区域和阶数方面都有很好的改进。为了检验这些方法与 SFPBM 和 MEBDF 方法相比在减少 CPU 时间、从而提高低阶和高阶新方法精度方面的优势,我们给出了一些数值示例。
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引用次数: 0
High-order schemes based on extrapolation for semilinear fractional differential equation 基于外推法的半线性分数微分方程高阶方案
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2023-12-11 DOI: 10.1007/s10092-023-00553-1
Yuhui Yang, Charles Wing Ho Green, Amiya K. Pani, Yubin Yan

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order (alpha in (1,2).) The error has the asymptotic expansion ( big ( d_{3} tau ^{3- alpha } + d_{4} tau ^{4-alpha } + d_{5} tau ^{5-alpha } + cdots big ) + big ( d_{2}^{*} tau ^{4} + d_{3}^{*} tau ^{6} + d_{4}^{*} tau ^{8} + cdots big ) ) at any fixed time (t_{N}= T, N in {mathbb {Z}}^{+}), where (d_{i}, i=3, 4,ldots ) and (d_{i}^{*}, i=2, 3,ldots ) denote some suitable constants and (tau = T/N) denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order (alpha in (1,2)) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

通过将黎曼-利奥维尔分数导数重写为哈达玛德有限部分积分,并借助片断二次插值多项式近似,建立了一个用于近似阶数为 (α in (1,2).) 的黎曼-利奥维尔分数导数的数值方案。误差具有渐近展开( big ( d_{3}tau ^{3- alpha }+ d_{4}tau ^{4-alpha }+ d_{5}tau ^{5-alpha }+ cdots big )+ big ( d_{2}^{*}tau ^{4}+ d_{3}^{*}tau ^{6}+ d_{4}^{*}tau ^{8}+ cdots big )),其中 (d_{i}, i=3, 4,ldots ) 和 (d_{i}^{*}, i=2, 3,ldots ) 表示一些合适的常数,而 (tau = T/N) 表示步长。在此离散化的基础上,推导出了一种新的近似线性分数微分方程的方案,其误差也有类似的渐近展开。因此,通过外推法得到了逼近线性分数微分方程的高阶方案。此外,还引入并分析了逼近半线性分数微分方程的高阶方案。我们进行了几次数值实验,结果表明数值结果与我们的理论发现是一致的。
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引用次数: 0
Explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation in the simultaneously massless and nonrelativistic regime 在同时无质量和非相对论条件下的狄拉克方程的显式和结构保持指数波积分器傅立叶伪谱方法
IF 1.7 2区 数学 Q1 MATHEMATICS Pub Date : 2023-12-11 DOI: 10.1007/s10092-023-00554-0
Jiyong Li

We propose two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral (SPEWIFP) methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. In this regime, the solution of Dirac equation is highly oscillatory in time because of the small parameter (0 <varepsilon ll 1) which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition (tau lesssim 1) and preserve the modified energy and modified mass in the discrete level. Although our methods can only preserve the modified energy and modified mass instead of the original energy and mass, our methods are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are often implicit. Through rigorous error analysis, we give the error bounds of the methods at (O(h^{m_0} + tau ^2/varepsilon ^2)) where h is mesh size, (tau ) is time step and the integer (m_0) is determined by the regularity conditions. These error bounds indicate that, to obtain the correct numerical solution in the simultaneously massless and nonrelativistic regime, our methods request the (varepsilon )-scalability as (h = O(1)) and (tau = O(varepsilon )) which is better than the (varepsilon )-scalability of the finite difference (FD) methods: (h =O(varepsilon ^{1/2})) and (tau = O(varepsilon ^{3/2})). Numerical experiments confirm that the theoretical results in this paper are correct.

我们提出了两种在同时无质量和非相对论条件下求解狄拉克方程的显式和结构保留指数波积分器傅立叶伪谱(SPEWIFP)方法。在这种情况下,由于与光速成反比的小参数(0 <varepsilon ll 1),狄拉克方程的解在时间上高度振荡。所提出的方法被证明是时间对称的,仅在 (tau lesssim 1) 条件下稳定,并在离散水平上保留修正能量和修正质量。虽然我们的方法只能保留修正能量和修正质量,而不能保留原始能量和质量,但我们的方法是显式的,与通常是隐式的传统结构保留方法相比,大大降低了计算成本。通过严格的误差分析,我们给出了方法的误差边界为 (O(h^{m_0} + tau ^2/varepsilon ^2)),其中 h 是网格大小,(tau )是时间步长,整数 (m_0)由正则条件决定。这些误差边界表明,为了在同时无质量和非相对论状态下获得正确的数值解,我们的方法要求具有 (varepsilon )-可扩展性,即 (h = O(1)) 和 (tau = O(varepsilon )) ,这比有限差分(FD)方法的 (varepsilon )-可扩展性要好:h =O(varepsilon ^{1/2})) and(tau = O(varepsilon ^{3/2})).数值实验证实了本文的理论结果是正确的。
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引用次数: 0
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Calcolo
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