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Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment 由内线段上的Cauchy数据重建半无限带中Helmholtz方程的辐射条件和求解
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-25 DOI: 10.1515/cmam-2022-0244
Pauline Achieng, F. Berntsson, V. Kozlov
Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.
摘要我们考虑亥姆霍兹方程的一个反问题,该反问题是根据对半无限带内的一段的测量来重建解。在条的两侧边界上规定了齐次Neumann条件,在边界的剩余部分上规定了未知的Dirichlet条件。额外的复杂性是无穷远处的辐射条件是未知的。我们的目的是找到Dirichlet边界条件和辐射条件下的未知函数。这样的问题出现在声学中,以根据声场测量来确定声源和表面振动。该问题分为两个子问题,一个是适定问题,另一个是不适定问题。我们分析了这两个问题的理论性质;特别地,我们证明了辐射条件是由一个稳定的非线性问题描述的。第二个问题是不适定的,我们使用Landweber迭代方法和差分原理对其进行正则化。数值测试表明,该方法效果良好。
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引用次数: 0
Recent Advances in Finite Element Methods 有限元方法的最新进展
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-25 DOI: 10.1515/cmam-2023-0135
Sven Beuchler, A. Rösch
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引用次数: 0
Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations 非发散型椭圆型偏微分方程的计算多尺度方法
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-24 DOI: 10.1515/cmam-2023-0040
Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler
This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.
本文提出了一种新的多尺度计算方法,用于求解满足Cordes条件的非散度非均质系数线性二阶椭圆型偏微分方程。该构造遵循定域正交分解(LOD)的方法,并在数值均匀化的精神下,通过在精细尺度上求解定域单元问题,提供适合算子的粗空间。粗糙空间的自由度与齐次问题的非协调和混合有限元方法有关。对一个示例性方法的严格误差分析表明,发散型偏微分方程中已知的LOD方法的有利性质,即其超越尺度分离和周期性的适用性和准确性,仍然适用于非发散形式的问题。
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引用次数: 0
Approximate Deconvolution with Correction – A High Fidelity Model for Magnetohydrodynamic Flows at High Reynolds and Magnetic Reynolds Numbers 带校正的近似反褶积——高雷诺数和磁雷诺数下磁流体动力流动的高保真模型
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-12 DOI: 10.1515/cmam-2022-0254
Yasasya Batugedara, A. Labovsky
Abstract We propose a model for magnetohydrodynamic flows at high Reynolds and magnetic Reynolds numbers. The system is written in the Elsässer variables so that the decoupling method of [C. Trenchea, Unconditional stability of a partitioned IMEX method for magnetohydrodynamic flows, Appl. Math. Lett. 27 (2014), 97–100] can be used. This decoupling method is only first-order accurate, so the proposed model aims at improving the temporal accuracy (from first to second order), as well as reducing the modeling error of the existing turbulence model. This is done in the framework of the recently developed LES-C turbulence models [A. E. Labovsky, Approximate deconvolution with correction: A member of a new class of models for high Reynolds number flows, SIAM J. Numer. Anal. 58 (2020), 5, 3068–3090]. We show the model to be unconditionally stable and numerically verify its superiority over its most natural competitor.
摘要我们提出了一个高雷诺数和磁雷诺数下的磁流体动力学流动模型。该系统以Elsässer变量编写,因此可以使用[C.Trenchea,磁流体动力学流的分区IMEX方法的无条件稳定性,Appl.Math.Lett.27(2014),97–100]的解耦方法。这种解耦方法只有一阶精度,因此所提出的模型旨在提高时间精度(从一阶到二阶),并降低现有湍流模型的建模误差。这是在最近开发的LES-C湍流模型[A的框架内完成的。 E.Labovsky,带校正的近似反褶积:一类新的高雷诺数流动模型的成员,SIAM J.Numer。Anal。58(2020),53068–3090]。我们证明了该模型是无条件稳定的,并在数值上验证了其相对于最自然竞争对手的优势。
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引用次数: 1
An Adaptive Two-Grid Solver for DPG Formulation of Compressible Navier–Stokes Equations in 3D 三维可压缩Navier-Stokes方程DPG格式的自适应双网格求解器
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-12 DOI: 10.1515/cmam-2022-0206
W. Rachowicz, W. Cecot, A. Zdunek
Abstract We present an overlapping domain decomposition iterative solver for linear systems resulting from the discretization of compressible viscous flows with the Discontinuous Petrov–Galerkin (DPG) method in three dimensions. It is a two-grid solver utilizing the solution on the auxiliary coarse grid and the standard block-Jacobi iteration on patches of elements defined by supports of the coarse mesh base shape functions. The simple iteration defined in this way is used as a preconditioner for the conjugate gradient procedure. Theoretical analysis indicates that the condition number of the preconditioned system should be independent of the actual finite element mesh and the auxiliary coarse mesh, provided that they are quasiuniform. Numerical tests confirm this result. Moreover, they show that presence of strongly flattened or elongated elements does not slow the convergence. The finite element mesh is subject to adaptivity, i.e. dividing the elements with large errors until a required accuracy is reached. The auxiliary coarse mesh is adjusting to the nonuniform actual mesh.
摘要我们提出了一个三维可压缩粘性流离散化线性系统的重叠域分解迭代求解器。它是一个双网格求解器,利用辅助粗网格上的解和由粗网格基本形状函数的支持定义的元素块上的标准块Jacobi迭代。以这种方式定义的简单迭代被用作共轭梯度过程的预处理器。理论分析表明,预处理系统的条件数应独立于实际有限元网格和辅助粗网格,前提是它们是拟均匀的。数值试验证实了这一结果。此外,它们表明,强扁平或细长元件的存在不会减缓收敛。有限元网格具有自适应性,即划分具有大误差的单元,直到达到所需的精度。辅助粗网格正在根据不均匀的实际网格进行调整。
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引用次数: 0
A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions 二维GDSW重叠Schwarz预条件的多级推广
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-12 DOI: 10.1515/cmam-2022-0168
Alexander Heinlein, O. Rheinbach, F. Röver
Abstract Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.
摘要本文考虑了广义Dryja–Smith–Widlund(GDSW)型重叠Schwarz域分解预条件子的多级扩张。最初的GDSW预处理器是一个两级重叠的Schwarz域分解预处理器,它可以由完全组装的刚度矩阵代数构造。FROSch软件属于Trilinos软件库的ShyLU包,它提供了GDSW预处理器不同变体的并行实现。粗糙问题会限制两级GDSW预处理器的并行可扩展性。作为补救措施,过去曾提出过三级GDSW方法,它可以显著扩展可扩展性的范围。本文介绍并分析了GDSW预处理器的多级扩展。最后,在FROSch中实现多达40个并行结果 000核的超级MUC NG超级计算机,以及48 000核的JUWELS超级计算机。
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引用次数: 0
Simultaneous Recovery of Two Time-Dependent Coefficients in a Multi-Term Time-Fractional Diffusion Equation 多项时间分数阶扩散方程中两个时间相关系数的同时恢复
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-12 DOI: 10.1515/cmam-2022-0210
Wenjun Ma, Liangliang Sun
Abstract This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.
摘要研究了多项时间分数扩散方程中两点测量数据同时确定时变势项和时变源函数的反问题。首先利用不动点定理研究了直接问题解的存在唯一性和一些规律。然后利用直接问题解的正则性和分数阶导数的优良性质,得到了反演系数问题的条件稳定性估计。此外,还说明了反问题的病态性,并将反问题转化为变分问题。此外,还给出了变分问题的最小值的存在性和收敛性。最后,利用改进的Levenberg-Marquardt方法对两个未知时相关系数的近似函数进行了数值重构。通过三个一维和二维实例的数值实验,验证了该方法的有效性和鲁棒性。
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引用次数: 0
Well-Posedness and Convergence Analysis of PML Method for Time-Dependent Acoustic Scattering Problems Over a Locally Rough Surface 局部粗糙表面时变声散射问题的PML方法的适定性和收敛性分析
IF 1.3 4区 数学 Q2 Mathematics Pub Date : 2023-07-07 DOI: 10.1515/cmam-2023-0017
Hongxia Guo, Guanghui Hu
We aim to analyze and calculate time-dependent acoustic wave scattering by a bounded obstacle and a locally perturbed non-selfintersecting curve. The scattering problem is equivalently reformulated as an initial-boundary value problem of the wave equation in a truncated bounded domain through a well-defined transparent boundary condition. Well-posedness and stability of the reduced problem are established. Numerically, we adopt the perfect matched layer (PML) scheme for simulating the propagation of perturbed waves. By designing a special absorbing medium in a semi-circular PML, we show the well-posedness and stability of the truncated initial-boundary value problem. Finally, we prove that the PML solution converges exponentially to the exact solution in the physical domain. Numerical results are reported to verify the exponential convergence with respect to absorbing medium parameters and thickness of the PML.
我们的目的是分析和计算声波在有界障碍物和局部摄动非自交曲线下的时变散射。通过明确定义的透明边界条件,将散射问题等效地转化为截断有界域中波动方程的初边值问题。建立了约简问题的适定性和稳定性。数值上,我们采用完全匹配层(PML)格式来模拟摄动波的传播。通过在半圆形PML中设计一种特殊的吸收介质,证明了截断初边值问题的适定性和稳定性。最后,我们证明了PML解在物理域中指数收敛于精确解。数值结果验证了吸波介质参数和吸波介质厚度的指数收敛性。
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引用次数: 0
Frontmatter 头版头条
4区 数学 Q2 Mathematics Pub Date : 2023-07-01 DOI: 10.1515/cmam-2023-frontmatter3
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引用次数: 0
A Domain Decomposition Scheme for Couplings between Local and Nonlocal Equations 局部与非局部方程耦合的一种区域分解格式
4区 数学 Q2 Mathematics Pub Date : 2023-06-20 DOI: 10.1515/cmam-2022-0140
Gabriel Acosta, Francisco M. Bersetche, Julio D. Rossi
Abstract We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions’s framework and prove, as a consequence, convergence in both the continuous and the discrete settings.
摘要研究了一类局部算子与非局部算子耦合的Schwarz型自然交替方法(域分解)。我们证明了我们的方法符合Lions的框架,并证明了在连续和离散设置下的收敛性。
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引用次数: 1
期刊
Computational Methods in Applied Mathematics
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