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A P-Version of Convolution Quadrature in Wave Propagation 波传播中卷积正交的p型
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-08-14 DOI: 10.1137/24m1642524
Alexander Rieder
{"title":"A P-Version of Convolution Quadrature in Wave Propagation","authors":"Alexander Rieder","doi":"10.1137/24m1642524","DOIUrl":"https://doi.org/10.1137/24m1642524","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1729-1756, August 2025. <br/> Abstract. We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the time step size ([math]-method), we achieve accuracy by increasing the order of the method ([math]-method). We base this method on discontinuous Galerkin time stepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observe a (root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the finding.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"105 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Improved High-Index Saddle Dynamics for Finding Saddle Points and Solution Landscape 改进的高指数鞍动态寻找鞍点和解决方案景观
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-08-14 DOI: 10.1137/25m173212x
Hua Su, Haoran Wang, Lei Zhang, Jin Zhao, Xiangcheng Zheng
{"title":"Improved High-Index Saddle Dynamics for Finding Saddle Points and Solution Landscape","authors":"Hua Su, Haoran Wang, Lei Zhang, Jin Zhao, Xiangcheng Zheng","doi":"10.1137/25m173212x","DOIUrl":"https://doi.org/10.1137/25m173212x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1757-1775, August 2025. <br/> Abstract. We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD and then prove its stable and nonlocal convergence from stationary points that may not be close to the target saddle point, which reduces the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD for implementation. Furthermore, based on Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Ideally, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144840296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
A Stochastic Preconditioned Douglas–Rachford Splitting Method for Saddle-Point Problems 鞍点问题的随机预条件Douglas-Rachford分裂方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-08-12 DOI: 10.1137/23m1622490
Yakun Dong, Kristian Bredies, Hongpeng Sun
{"title":"A Stochastic Preconditioned Douglas–Rachford Splitting Method for Saddle-Point Problems","authors":"Yakun Dong, Kristian Bredies, Hongpeng Sun","doi":"10.1137/23m1622490","DOIUrl":"https://doi.org/10.1137/23m1622490","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1691-1728, August 2025. <br/> Abstract. In this article, we propose and study a stochastic and relaxed preconditioned Douglas–Rachford splitting method to solve saddle-point problems that have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convex-concave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence concerning the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic and relaxed preconditioned Douglas–Rachford splitting methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144819998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Quasi-Monte Carlo for Partial Differential Equations with Generalized Gaussian Input Uncertainty 广义高斯输入不确定性偏微分方程的拟蒙特卡罗算法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-08-11 DOI: 10.1137/24m1708164
Philipp A. Guth, Vesa Kaarnioja
{"title":"Quasi-Monte Carlo for Partial Differential Equations with Generalized Gaussian Input Uncertainty","authors":"Philipp A. Guth, Vesa Kaarnioja","doi":"10.1137/24m1708164","DOIUrl":"https://doi.org/10.1137/24m1708164","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1666-1690, August 2025. <br/> Abstract. There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Lê [Comput. Math. Appl., 164 (2024), pp. 116–130; SIAM J. Numer. Anal., 62 (2024), pp. 1874–1900] as well as Harbrecht, Schmidlin, and Schwab [Math. Models Methods Appl. Sci., 34 (2024), pp. 881–917] analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"42 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144819942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
A Stabilized Nonconforming Finite Element Method for the Surface Biharmonic Problem 表面双调和问题的稳定非协调有限元法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-08-06 DOI: 10.1137/24m1707936
Shuonan Wu, Hao Zhou
{"title":"A Stabilized Nonconforming Finite Element Method for the Surface Biharmonic Problem","authors":"Shuonan Wu, Hao Zhou","doi":"10.1137/24m1707936","DOIUrl":"https://doi.org/10.1137/24m1707936","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1642-1665, August 2025. <br/> Abstract. This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its [math]-relative conformity and weak [math] conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only [math] regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken [math] norm. Numerical experiments are provided to support the theoretical results and indicate that the stabilization term might be unnecessary.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144787661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
A Localized Orthogonal Decomposition Method for Heterogeneous Stokes Problems 非均质Stokes问题的局部正交分解方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-08-01 DOI: 10.1137/24m1704166
Moritz Hauck, Alexei Lozinski
{"title":"A Localized Orthogonal Decomposition Method for Heterogeneous Stokes Problems","authors":"Moritz Hauck, Alexei Lozinski","doi":"10.1137/24m1704166","DOIUrl":"https://doi.org/10.1137/24m1704166","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1617-1641, August 2025. <br/> Abstract. In this paper, we propose a multiscale method for heterogeneous Stokes problems. The method is based on the localized orthogonal decomposition (LOD) methodology and has approximation properties independent of the regularity of the coefficients. We apply the LOD to an appropriate reformulation of the Stokes problem, which allows us to construct exponentially decaying basis functions for the velocity approximation while using a piecewise constant pressure approximation. The exponential decay motivates a localization of the basis computation, which is essential for the practical realization of the method. We perform a rigorous a priori error analysis and prove optimal convergence rates for the velocity approximation and a postprocessed pressure approximation, provided that the supports of the basis functions are logarithmically increased with the desired accuracy. Numerical experiments support the theoretical results of this paper.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"69 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144766090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Error Analysis of BDF 1–6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates BDF - 6时间步进方法在瞬态斯托克斯问题中的误差分析：速度和压力估计
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-24 DOI: 10.1137/23m1606800
Alessandro Contri, Balázs Kovács, André Massing
{"title":"Error Analysis of BDF 1–6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates","authors":"Alessandro Contri, Balázs Kovács, André Massing","doi":"10.1137/23m1606800","DOIUrl":"https://doi.org/10.1137/23m1606800","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1586-1616, August 2025. <br/> Abstract. We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [SIAM J. Numer. Anal., 47 (2009), pp. 409–439] and provide a unified theoretical analysis of backward difference formula methods of orders 1 to 6. The main novelty of our approach lies in deriving optimal-order stability and error estimates for both the velocity and the pressure using Dahlquist’s [math]-stability concept together with the multiplier technique introduced by Nevanlinna and Odeh and recently by Akrivis et al. [SIAM J. Numer. Anal., 59 (2021), pp. 2449–2472]. When combined with a method-dependent Ritz projection for the initial data, unconditional stability can be shown, while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"119 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Trefftz Discontinuous Galerkin Approximation of an Acoustic Waveguide 声波导的Trefftz不连续伽辽金近似
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-21 DOI: 10.1137/24m1686905
Peter Monk, Manuel Pena, Virginia Selgas
{"title":"Trefftz Discontinuous Galerkin Approximation of an Acoustic Waveguide","authors":"Peter Monk, Manuel Pena, Virginia Selgas","doi":"10.1137/24m1686905","DOIUrl":"https://doi.org/10.1137/24m1686905","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1561-1585, August 2025. <br/> Abstract. We propose a modified Trefftz discontinuous Galerkin (TDG) method for approximating a time-harmonic acoustic scattering problem in an infinitely elongated waveguide. In the waveguide we suppose that there is a bounded, penetrable, and possibly absorbing scatterer. The classical TDG is not applicable when the scatterer is absorbing. Novel features of our modified TDG method are that it is applicable in this case, and it uses a stable treatment of the outgoing radiation condition for the scattered field. For the modified TDG, we prove [math] and [math]-convergence in the [math] norm for nonabsorbing scatterers. The theoretical results are verified numerically for a discretization based on plane waves, and also investigated numerically for absorbing scatterers (in which case the plane waves are evanescent in the scatterer).","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144669721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
A Posteriori Error Control for the Allen–Cahn Equation with Variable Mobility 变迁移率Allen-Cahn方程的后验误差控制
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-17 DOI: 10.1137/24m1646406
A. Brunk, J. Giesselmann, M. Lukáčová-Medvi[math]ová
{"title":"A Posteriori Error Control for the Allen–Cahn Equation with Variable Mobility","authors":"A. Brunk, J. Giesselmann, M. Lukáčová-Medvi[math]ová","doi":"10.1137/24m1646406","DOIUrl":"https://doi.org/10.1137/24m1646406","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1540-1560, August 2025. <br/> Abstract. In this work, we derive a [math]-robust a posteriori error estimator for finite element approximations of the Allen–Cahn equation with variable nondegenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144645433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion 拟线性次扩散的正则性分析及高阶时间步进格式
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-17 DOI: 10.1137/23m159531x
Bangti Jin, Qimeng Quan, Barbara Wohlmuth, Zhi Zhou
{"title":"Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion","authors":"Bangti Jin, Qimeng Quan, Barbara Wohlmuth, Zhi Zhou","doi":"10.1137/23m159531x","DOIUrl":"https://doi.org/10.1137/23m159531x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1512-1539, August 2025. <br/> Abstract. In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order [math] in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several new pointwise-in-time Sobolev regularity estimates that are useful for numerical analysis. Then we develop a time-stepping scheme to solve quasilinear subdiffusion, based on convolution quadrature generated by the second-order backward differentiation formula with a correction at the first step. Further, we establish that the convergence order of the scheme is [math] without imposing any additional assumption on the regularity of the solution, which is high-order in the sense that its convergence rate is higher than the first-order convergence of the vanilla scheme. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions are presented to show the sharpness of the error estimate.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"84 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144645495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
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