SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2395-2420, December 2025. Abstract. We establish guaranteed and practically computable a posteriori error bounds for source problems and eigenvalue problems involving linear Schrödinger operators with atom-centered potentials discretized with linear combinations of atomic orbitals. We show that the energy norm of the discretization error can be estimated by the dual energy norm of the residual, that further decomposes into atomic contributions, characterizing the error localized on atoms. Moreover, we show that the practical computation of the dual norms of atomic residuals involves diagonalizing radial Schrödinger operators which can easily be precomputed in practice. We provide numerical illustrations of the performance of such a posteriori analysis on several test cases, showing that the error bounds accurately estimate the error, and that the localized error components allow for optimized adaptive basis sets.
{"title":"A Posteriori Error Estimates for Schrödinger Operators Discretized with Linear Combinations of Atomic Orbitals","authors":"Mi-Song Dupuy, Geneviève Dusson, Ioanna-Maria Lygatsika","doi":"10.1137/24m1700697","DOIUrl":"https://doi.org/10.1137/24m1700697","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2395-2420, December 2025. <br/> Abstract. We establish guaranteed and practically computable a posteriori error bounds for source problems and eigenvalue problems involving linear Schrödinger operators with atom-centered potentials discretized with linear combinations of atomic orbitals. We show that the energy norm of the discretization error can be estimated by the dual energy norm of the residual, that further decomposes into atomic contributions, characterizing the error localized on atoms. Moreover, we show that the practical computation of the dual norms of atomic residuals involves diagonalizing radial Schrödinger operators which can easily be precomputed in practice. We provide numerical illustrations of the performance of such a posteriori analysis on several test cases, showing that the error bounds accurately estimate the error, and that the localized error components allow for optimized adaptive basis sets.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"163 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145765628","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 2371-2394, December 2025. Abstract. Signature kernels have become a powerful tool in kernel methods for sequential data. In “The Signature Kernel is the solution of a Goursat PDE” [], the authors introduced a kernel trick showing that, for continuously differentiable paths, the signature kernel satisfies a hyperbolic PDE of Goursat type in two independent time variables. While finite difference methods have been explored for this PDE, they suffer from accuracy and stability issues when handling highly oscillatory inputs. In this work, we propose two advanced numerical schemes that approximate the solution using polynomial representations of boundary conditions and employing either approximation or interpolation techniques. We prove the convergence of the polynomial approximation scheme and demonstrate experimentally that both methods achieve several orders of magnitude improvement in mean absolute percentage error (MAPE) over finite difference schemes without increasing computational complexity. These algorithms are implemented in a publicly available Python library: https://github.com/FrancescoPiatti/polysigkernel.
SIAM数值分析杂志，第63卷，第6期，2371-2394页，2025年12月。摘要。在序列数据核方法中，签名核已经成为一种强大的工具。在“The Signature Kernel is a Goursat PDE的解”[]中，作者介绍了一个核技巧，表明对于连续可微路径，签名核满足两个独立时间变量的Goursat型双曲PDE。虽然有限差分方法已经探索了这种PDE，但它们在处理高振荡输入时存在精度和稳定性问题。在这项工作中，我们提出了两种先进的数值方案，使用边界条件的多项式表示和采用近似或插值技术来近似解。我们证明了多项式近似格式的收敛性，并通过实验证明了这两种方法在不增加计算复杂度的情况下，比有限差分格式在平均绝对百分比误差（MAPE）方面取得了几个数量级的改进。这些算法在一个公开可用的Python库中实现：https://github.com/FrancescoPiatti/polysigkernel。
{"title":"Numerical Schemes for Signature Kernels","authors":"Thomas Cass, Francesco Piatti, Jeffrey Pei","doi":"10.1137/25m1740681","DOIUrl":"https://doi.org/10.1137/25m1740681","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2371-2394, December 2025. <br/> Abstract. Signature kernels have become a powerful tool in kernel methods for sequential data. In “The Signature Kernel is the solution of a Goursat PDE” [], the authors introduced a kernel trick showing that, for continuously differentiable paths, the signature kernel satisfies a hyperbolic PDE of Goursat type in two independent time variables. While finite difference methods have been explored for this PDE, they suffer from accuracy and stability issues when handling highly oscillatory inputs. In this work, we propose two advanced numerical schemes that approximate the solution using polynomial representations of boundary conditions and employing either approximation or interpolation techniques. We prove the convergence of the polynomial approximation scheme and demonstrate experimentally that both methods achieve several orders of magnitude improvement in mean absolute percentage error (MAPE) over finite difference schemes without increasing computational complexity. These algorithms are implemented in a publicly available Python library: https://github.com/FrancescoPiatti/polysigkernel.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145718456","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 2343-2370, December 2025. Abstract. A geometric perspective on the gentlest ascent dynamics is presented, revealing that the dynamics is utilizing the Householder reflector—constructed via the continuous power method—to adapt the negative gradient and identify index-1 saddle points. While the adaptation appears intuitive, it is governed by a precise criterion. Building on this geometric insight, three generalized dynamical systems are introduced for locating high-index saddle points, each centered on estimating directions for constructing generalized reflectors. The first approach employs the Oja flow to evolve eigenspaces, encompassing the continuous power method as a special case. The second approach formulates a matrix Riccati differential equation for the projector operator on the Grassmann manifold, which is shown to be equivalent to a double bracket flow with inherent sorting properties. The third approach is a hybrid method based on conventional subspace iteration, incorporating [math] factorization for normalization. The equilibrium points of all three systems are classified, and convergence analyses are provided. These dynamical systems are readily solvable by using high-precision numerical ODE integrators. Numerical experiments confirm the theoretical results.
{"title":"Generalized Gentlest Ascent Dynamics Methods for High-Index Saddle Points","authors":"Moody T. Chu, Matthew M. Lin","doi":"10.1137/24m1710905","DOIUrl":"https://doi.org/10.1137/24m1710905","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2343-2370, December 2025. <br/> Abstract. A geometric perspective on the gentlest ascent dynamics is presented, revealing that the dynamics is utilizing the Householder reflector—constructed via the continuous power method—to adapt the negative gradient and identify index-1 saddle points. While the adaptation appears intuitive, it is governed by a precise criterion. Building on this geometric insight, three generalized dynamical systems are introduced for locating high-index saddle points, each centered on estimating directions for constructing generalized reflectors. The first approach employs the Oja flow to evolve eigenspaces, encompassing the continuous power method as a special case. The second approach formulates a matrix Riccati differential equation for the projector operator on the Grassmann manifold, which is shown to be equivalent to a double bracket flow with inherent sorting properties. The third approach is a hybrid method based on conventional subspace iteration, incorporating [math] factorization for normalization. The equilibrium points of all three systems are classified, and convergence analyses are provided. These dynamical systems are readily solvable by using high-precision numerical ODE integrators. Numerical experiments confirm the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145559903","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}
L. Beirão da Veiga, D. A. Di Pietro, J. Droniou, K. B. Haile, T. J. Radley
SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2317-2342, December 2025. Abstract. In this paper we propose and analyze a new finite element method for the solution of the two- and three-dimensional incompressible Navier–Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting preasymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments.
{"title":"A Reynolds-Semirobust Method with Hybrid Velocity and Pressure for the Unsteady Incompressible Navier–Stokes Equations","authors":"L. Beirão da Veiga, D. A. Di Pietro, J. Droniou, K. B. Haile, T. J. Radley","doi":"10.1137/25m1736104","DOIUrl":"https://doi.org/10.1137/25m1736104","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2317-2342, December 2025. <br/> Abstract. In this paper we propose and analyze a new finite element method for the solution of the two- and three-dimensional incompressible Navier–Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting preasymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145536093","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 2296-2316, December 2025. Abstract. We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous, preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.
{"title":"Boundary-Value Problems of Functional Differential Equations with State-Dependent Delays","authors":"Alessia Andò, Jan Sieber","doi":"10.1137/24m1711182","DOIUrl":"https://doi.org/10.1137/24m1711182","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2296-2316, December 2025. <br/> Abstract. We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous, preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"81 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145485828","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 2272-2295, December 2025. Abstract. The quad-curl problem is a critical issue in magnetohydrodynamics and inverse electromagnetic scattering theory. It has traditionally been addressed by most existing numerical schemes through the formation of saddle-point systems, thereby introducing substantial challenges for both theoretical analysis and practical numerical implementations. This study introduces a novel regularization-based approach that diverges from these conventional methods, specifically designed to avoid the saddle-point issue. The challenge of addressing the divergence-free constraint in finite element methods is tackled in a unique way. Moreover, it ensures a consistent well-posedness, leading to a symmetric, positive-definite system in finite element discretization, which simplifies the implementation process. The regularized problem is addressed using the conforming finite element method, employing [math]-conforming element, and the discontinuous Galerkin method, utilizing Nédélec’s element, both of which achieve quasi-optimal error bounds in relevant norms. The efficiency of our proposed methods is further demonstrated through a series of numerical experiments in both two and three dimensions.
{"title":"An Efficient Finite Element Method for the Quad-Curl Problem","authors":"Jingzhi Li, Shipeng Mao, Chao Wang, Zhimin Zhang","doi":"10.1137/24m166022x","DOIUrl":"https://doi.org/10.1137/24m166022x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2272-2295, December 2025. <br/> Abstract. The quad-curl problem is a critical issue in magnetohydrodynamics and inverse electromagnetic scattering theory. It has traditionally been addressed by most existing numerical schemes through the formation of saddle-point systems, thereby introducing substantial challenges for both theoretical analysis and practical numerical implementations. This study introduces a novel regularization-based approach that diverges from these conventional methods, specifically designed to avoid the saddle-point issue. The challenge of addressing the divergence-free constraint in finite element methods is tackled in a unique way. Moreover, it ensures a consistent well-posedness, leading to a symmetric, positive-definite system in finite element discretization, which simplifies the implementation process. The regularized problem is addressed using the conforming finite element method, employing [math]-conforming element, and the discontinuous Galerkin method, utilizing Nédélec’s element, both of which achieve quasi-optimal error bounds in relevant norms. The efficiency of our proposed methods is further demonstrated through a series of numerical experiments in both two and three dimensions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"114 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145462322","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 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}
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