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Geometric Low-Rank Approximation of the Zeitlin Model of Incompressible Fluids on the Sphere 球面上不可压缩流体的Zeitlin模型的几何低秩逼近
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-11-04 DOI: 10.1137/24m1718925
Cecilia Pagliantini
{"title":"Geometric Low-Rank Approximation of the Zeitlin Model of Incompressible Fluids on the Sphere","authors":"Cecilia Pagliantini","doi":"10.1137/24m1718925","DOIUrl":"https://doi.org/10.1137/24m1718925","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2221-2248, December 2025. <br/> Abstract. We consider the vorticity formulation of the Euler equations describing the flow of a two-dimensional incompressible ideal fluid on the sphere. Zeitlin’s model provides a finite-dimensional approximation of the vorticity formulation that preserves the underlying geometric structure: it consists of an isospectral Lie–Poisson flow on the Lie algebra of skew-Hermitian matrices. We propose an approximation of Zeitlin’s model based on a time-dependent low-rank factorization of the vorticity matrix and evolve a basis of eigenvectors according to the Euler equations. In particular, we show that the approximate flow remains isospectral and Lie–Poisson and that the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions, only depends on the approximation of the vorticity matrix at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly if a further approximation of the stream function is introduced.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145434055","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
Two-Level Hybrid Schwarz Preconditioners for the Helmholtz Equation with High Wave Number 高波数Helmholtz方程的两能级混合Schwarz预条件
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-11-03 DOI: 10.1137/24m168533x
Peipei Lu, Xuejun Xu, Bowen Zheng, Jun Zou
{"title":"Two-Level Hybrid Schwarz Preconditioners for the Helmholtz Equation with High Wave Number","authors":"Peipei Lu, Xuejun Xu, Bowen Zheng, Jun Zou","doi":"10.1137/24m168533x","DOIUrl":"https://doi.org/10.1137/24m168533x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2187-2220, December 2025. <br/> Abstract. In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique, which was proposed by Målqvist and Peterseim [Math. Comp., 83 (2014), pp. 2583–2603] and Peterseim [Math. Comp., 86 (2017), pp. 1005–1036] originally for the discretization schemes for elliptic multiscale problems with heterogeneous and highly oscillating coefficients and Helmholtz problems with high wave number to eliminate the pollution effect. The local subproblems are Helmholtz problems in subdomains with homogeneous boundary conditions (the first preconditioner) or impedance boundary conditions (the second preconditioner). Both preconditioners are shown to be optimal under reasonable conditions; that is, a uniform upper bound of the preconditioned operator norm and a uniform lower bound of the field of values are established in terms of all the key parameters, such as fine mesh size, coarse mesh size, subdomain size, and wave numbers. This is the first rigorous demonstration of the optimality of a two-level Schwarz-type method with respect to all the key parameters and of the fact that the LOD solver can be a very effective coarse solver when it is used appropriately in the Schwarz method with multiple overlapping subdomains for the Helmholtz equation with high wave number in both two and three dimensions. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed preconditioners.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"28 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145427831","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 Priori and A Posteriori Error Identities for the Scalar Signorini Problem 标量Signorini问题的先验和后验误差恒等式
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-10-16 DOI: 10.1137/24m1677691
Sören Bartels, Thirupathi Gudi, Alex Kaltenbach
{"title":"A Priori and A Posteriori Error Identities for the Scalar Signorini Problem","authors":"Sören Bartels, Thirupathi Gudi, Alex Kaltenbach","doi":"10.1137/24m1677691","DOIUrl":"https://doi.org/10.1137/24m1677691","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2155-2186, October 2025. <br/> Abstract. In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an a posteriori error identity for arbitrary conforming approximations of the primal formulation and the dual formulation of the scalar Signorini problem. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an a priori error identity that applies to the approximation of the primal formulation using the Crouzeix–Raviart element and to the approximation of the dual formulation using the Raviart–Thomas element, and leads to quasi-optimal error decay rates without imposing additional assumptions on the contact set and in arbitrary space dimensions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"91 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145310706","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
Convergence of the Dirichlet–Neumann Alternating Method for Semilinear Elliptic Equations 半线性椭圆方程的Dirichlet-Neumann交替方法的收敛性
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-10-14 DOI: 10.1137/24m1703550
Emil Engström
{"title":"Convergence of the Dirichlet–Neumann Alternating Method for Semilinear Elliptic Equations","authors":"Emil Engström","doi":"10.1137/24m1703550","DOIUrl":"https://doi.org/10.1137/24m1703550","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2133-2154, October 2025. <br/> Abstract. The Dirichlet–Neumann alternating method is a common domain decomposition method for nonoverlapping domain decompositions without cross-points, and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet–Neumann alternating method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov–Poincaré formulation of the Dirichlet–Neumann alternating method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"58 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145289265","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
An Extension of the Euler–Maclaurin Summation Formula to Functions with Near Singularity 欧拉-麦克劳林求和公式在近奇异函数上的推广
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-10-13 DOI: 10.1137/24m1697530
Bowei Wu
{"title":"An Extension of the Euler–Maclaurin Summation Formula to Functions with Near Singularity","authors":"Bowei Wu","doi":"10.1137/24m1697530","DOIUrl":"https://doi.org/10.1137/24m1697530","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2119-2132, October 2025. <br/> Abstract. An extension of the Euler–Maclaurin (E–M) formula to near-singular functions is presented. This extension is derived based on earlier generalized E–M formulas for singular functions. The new E–M formulas consist of two components: a “singular” component that is a continuous extension of the earlier singular E–M formulas, and a “jump” component associated with the discontinuity of the integral with respect to a parameter that controls near singularity. The singular component of the new E–M formulas is an asymptotic series whose coefficients depend on the Hurwitz zeta function or the digamma function. Numerical examples of near-singular quadrature based on the extended E–M formula are presented, where accuracies of machine precision are achieved insensitive to the strength of the near singularity and with a very small number of quadrature nodes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"137 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145277500","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
Sensitivity of ODE Solutions and Quantities of Interest with Respect to Component Functions in the Dynamics 动力学中关于分量函数的ODE解的灵敏度和感兴趣的量
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-10-09 DOI: 10.1137/25m1729563
Jonathan R. Cangelosi, Matthias Heinkenschloss
{"title":"Sensitivity of ODE Solutions and Quantities of Interest with Respect to Component Functions in the Dynamics","authors":"Jonathan R. Cangelosi, Matthias Heinkenschloss","doi":"10.1137/25m1729563","DOIUrl":"https://doi.org/10.1137/25m1729563","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2094-2118, October 2025. <br/> Abstract. This work analyzes the sensitivities of the solution of a system of ordinary differential equations (ODEs) and a corresponding quantity of interest (QoI) to perturbations in a state-dependent component function that appears in the governing ODEs. This extends existing ODE sensitivity results, which consider the sensitivity of the ODE solution with respect to state-independent parameters. It is shown that with Carathéodory-type assumptions on the ODEs, the implicit function theorem can be applied to establish continuous Fréchet differentiability of the ODE solution with respect to the component function. These sensitivities are used to develop new estimates for the change in the ODE solution or QoI when the component function is perturbed. In applications, this new sensitivity-based bound on the ODE solution or QoI error is often much tighter than classical Gronwall-type error bounds. The sensitivity-based error bounds are applied to a trajectory simulation for a hypersonic vehicle.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145246967","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
Spherical Zone t-Designs for Numerical Integration and Approximation 数值积分与逼近的球面t区设计
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-09-26 DOI: 10.1137/24m1718883
Chao Li, Xiaojun Chen
{"title":"Spherical Zone t-Designs for Numerical Integration and Approximation","authors":"Chao Li, Xiaojun Chen","doi":"10.1137/24m1718883","DOIUrl":"https://doi.org/10.1137/24m1718883","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2072-2093, October 2025. <br/> Abstract. In this paper, we present spherical zone [math]-designs, which provide quadrature rules with equal weight for spherical polynomials of degree at most [math] on a spherical zone [math] with [math] and [math]. The spherical zone [math]-design is constructed by combining spherical [math]-designs and trapezoidal rules on [math] with polynomial exactness [math]. We show that the spherical zone [math]-designs using spherical [math]-designs only provide quadrature rules with equal weight for spherical zonal polynomials of degree at most [math] on the spherical zone. We apply the proposed spherical zone [math]-designs to numerical integration, hyperinterpolation and sparse approximation on the spherical zone. Theoretical approximation error bounds are presented. Some numerical examples are given to illustrate the theoretical results and show the efficiency of the proposed spherical zone [math]-designs.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145154125","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
Stability and Convergence of HDG Schemes under Minimal Regularity 最小正则性条件下HDG格式的稳定性和收敛性
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-09-25 DOI: 10.1137/23m1612846
Jiannan Jiang, Noel J. Walkington, Yukun Yue
{"title":"Stability and Convergence of HDG Schemes under Minimal Regularity","authors":"Jiannan Jiang, Noel J. Walkington, Yukun Yue","doi":"10.1137/23m1612846","DOIUrl":"https://doi.org/10.1137/23m1612846","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2048-2071, October 2025. <br/> Abstract. Convergence and compactness properties of approximate solutions to elliptic partial differential equations computed with the hybridized discontinuous Galerkin (HDG) scheme of Cockburn, Gopalakrishnan, and Sayas (Math. Comp., 79 (2010), pp. 1351–1367) are established. While it is known that solutions computed using this scheme converge at optimal rates to smooth solutions, this does not establish the stability of the method or convergence to solutions with minimal regularity. The compactness and convergence results show that the HDG scheme can be utilized for the solution of nonlinear problems and linear problems with nonsmooth coefficients on domains with reentrant corners.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145141191","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
Approximating Volumetric Shape Gradients for Shape Optimization with Curved Boundaries Constrained by Parabolic PDEs 抛物线偏微分方程约束下曲面边界形状优化的近似体积形状梯度
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-09-23 DOI: 10.1137/24m1681938
Leonardo Mutti, Michael Ulbrich
{"title":"Approximating Volumetric Shape Gradients for Shape Optimization with Curved Boundaries Constrained by Parabolic PDEs","authors":"Leonardo Mutti, Michael Ulbrich","doi":"10.1137/24m1681938","DOIUrl":"https://doi.org/10.1137/24m1681938","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2026-2047, October 2025. <br/> Abstract. We quantify the accuracy of the approximate shape gradient for a shape optimization problem constrained by parabolic PDEs. The focus is on the volume form of the shape gradient, which is discretized using the finite element method and the implicit Euler scheme. Our estimate goes beyond previous work done in the elliptic setting and considers the error introduced by polygonal approximation of curved domains. Numerical experiments support the theoretical findings, and the code is made publicly available.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"156 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145127788","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
On the Coefficients in Finite Difference Series Expansions of Derivatives 导数有限差分级数展开式中的系数
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-09-18 DOI: 10.1137/25m1731782
J. W. Banks, W. D. Henshaw
{"title":"On the Coefficients in Finite Difference Series Expansions of Derivatives","authors":"J. W. Banks, W. D. Henshaw","doi":"10.1137/25m1731782","DOIUrl":"https://doi.org/10.1137/25m1731782","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2009-2025, October 2025. <br/> Abstract. The formulation of finite difference approximations is a classical problem in numerical analysis. In this article, we consider difference approximations that are based on a series expansion in powers of the second undivided difference. Each additional term in the series increases the order of accuracy by two. These expansions are useful in a variety of contexts such as in the development of modified equation schemes, the design of high-order accurate energy stable discretizations, and error analysis of certain finite element or finite difference schemes. Here, we provide closed form expressions for the coefficients in the series expansions for derivatives of all orders. We also provide some short recursions defining the series coefficients, and formulae for the stencil coefficients in standard difference approximations. The series expansions are used to show some useful properties of the Fourier symbols of difference approximations and to derive rules of thumb for the number of points-per-wavelength needed to achieve a given error tolerance when solving wave propagation problems involving higher spatial derivatives.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"79 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145083733","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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