SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2249-2271, December 2025. Abstract. Lawson’s iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson’s iteration for the rational minimax approximation problem with both computationally high efficiency and theoretical guarantee is challenging. The recent work [L.-H. Zhang et al., Math. Comp., 94 (2025), pp. 2457–2494] reveals that Lawson’s iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation problem, and the work proposes a new type of Lawson’s iteration, namely, d-Lawson, which reduces to the classical Lawson’s iteration for the linear minimax approximation problem. For the rational case, such a dual problem is guaranteed to obtain the original minimax solution under Ruttan’s sufficient condition, and, numerically, d-Lawson was observed to converge monotonically with respect to the dual objective function. In this paper, we present a theoretical convergence analysis of d-Lawson for both the linear and rational minimax approximation problems. In particular, we show that (i) for the linear minimax approximation problem, [math] is a near-optimal Lawson exponent in Lawson’s iteration; and (ii) for the rational minimax approximation problem, under certain conditions, d-Lawson converges monotonically with respect to the dual objective function for any sufficiently small [math], and the limiting approximant satisfies the complementary slackness condition which states that any node associated with positive weight is either an interpolation point or has a constant error.
SIAM数值分析杂志，第63卷，第6期，2249-2271页，2025年12月。摘要。Lawson迭代法是求解复平面线性（多项式）极大极小逼近问题的一种经典而有效的方法。对有理极大极小逼近问题的劳森迭代法的推广具有较高的计算效率和理论保证。最近的工作[l . h .]Zhang et al.，数学。Comp., 94 (2025), pp. 2457-2494]揭示了Lawson迭代可以看作是解决原有理极大极小逼近问题对偶问题的一种方法，并提出了一种新型的Lawson迭代，即d-Lawson，将其简化为线性极大极小逼近问题的经典Lawson迭代。对于有理情况，在Ruttan的充分条件下，保证了该对偶问题得到原始的极大极小解，并且在数值上观察到d-Lawson对对偶目标函数的单调收敛。本文对线性极大极小逼近问题和有理极大极小逼近问题给出了d-Lawson的理论收敛性分析。特别地，我们证明了(i)对于线性极大极小逼近问题，[math]是Lawson迭代中的近最优Lawson指数；（ii）对于有理极大极小逼近问题，在一定条件下，对于任何足够小的对偶目标函数d-Lawson单调收敛[math]，并且极限逼近满足互补松弛条件，即任何与正权相关的节点要么是插值点，要么具有恒定误差。
{"title":"A Convergence Analysis Of Lawson’s Iteration For Computing Polynomial And Rational Minimax Approximations","authors":"Lei-Hong Zhang, Shanheng Han","doi":"10.1137/24m1708814","DOIUrl":"https://doi.org/10.1137/24m1708814","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2249-2271, December 2025. <br/> Abstract. Lawson’s iteration is a classical and effective method for solving the linear (polynomial) minimax approximation problem in the complex plane. Extension of Lawson’s iteration for the rational minimax approximation problem with both computationally high efficiency and theoretical guarantee is challenging. The recent work [L.-H. Zhang et al., Math. Comp., 94 (2025), pp. 2457–2494] reveals that Lawson’s iteration can be viewed as a method for solving the dual problem of the original rational minimax approximation problem, and the work proposes a new type of Lawson’s iteration, namely, d-Lawson, which reduces to the classical Lawson’s iteration for the linear minimax approximation problem. For the rational case, such a dual problem is guaranteed to obtain the original minimax solution under Ruttan’s sufficient condition, and, numerically, d-Lawson was observed to converge monotonically with respect to the dual objective function. In this paper, we present a theoretical convergence analysis of d-Lawson for both the linear and rational minimax approximation problems. In particular, we show that (i) for the linear minimax approximation problem, [math] is a near-optimal Lawson exponent in Lawson’s iteration; and (ii) for the rational minimax approximation problem, under certain conditions, d-Lawson converges monotonically with respect to the dual objective function for any sufficiently small [math], and the limiting approximant satisfies the complementary slackness condition which states that any node associated with positive weight is either an interpolation point or has a constant error.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145455188","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2221-2248, December 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2187-2220, December 2025. 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.
SIAM数值分析杂志，63卷，第6期，2187-2220页，2025年12月。摘要。本文提出并分析了求解二维和三维高波数亥姆霍兹方程的两能级混合Schwarz预条件。这两个预条件都定义在一组重叠的子域上，每个预条件由每个子域上的一个全局粗解器和一个局部求解器组成。全局粗解器基于ma lqvist和Peterseim [Math]提出的局部正交分解（LOD）技术。比较，83 (2014),pp. 2583-2603[数学]。Comp., 86 (2017), pp. 1005-1036]最初用于非均匀和高振荡系数椭圆多尺度问题和高波数Helmholtz问题的离散化方案，以消除污染效应。局部子问题是具有齐次边界条件（第一预条件）或阻抗边界条件（第二预条件）的子域上的Helmholtz问题。在合理条件下，两种预调节器均为最优；即根据细网格大小、粗网格大小、子域大小、波数等关键参数，建立了预置算子范数的均匀上界和值域的均匀下界。这是关于所有关键参数的两级Schwarz型方法的最优性的第一个严格证明，并且当LOD求解器在二维和三维具有高波数的亥姆霍兹方程的具有多个重叠子域的Schwarz方法中适当使用时，它可以是一个非常有效的粗求解器。通过数值实验验证了这两种预调节器的最优性和效率。
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2155-2186, October 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2133-2154, October 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2119-2132, October 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2094-2118, October 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2072-2093, October 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2048-2071, October 2025. 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.
SIAM数值分析杂志，第63卷，第5期，2048-2071页，2025年10月。摘要。用Cockburn， Gopalakrishnan和Sayas的杂交不连续Galerkin （HDG）格式计算椭圆型偏微分方程近似解的收敛性和紧性。Comp., 79 (2010), pp. 1351-1367)建立。虽然已知使用该格式计算的解以最优速率收敛到光滑解，但这并不能确定该方法的稳定性或收敛到具有最小规则性的解。紧凑性和收敛性结果表明，HDG格式可用于求解具有可重入角域上的非线性和非光滑系数线性问题。
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 5, Page 2026-2047, October 2025. 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.
{"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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号