SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 148-169, February 2026. Abstract. In this paper, we propose and analyze a product integration method for the second-kind integral equation with weakly singular and continuous kernels on the unit sphere [math]. We employ quadrature rules that satisfy the Marcinkiewicz–Zygmund property to construct hyperinterpolation for approximating the product of the continuous kernel and the solution in terms of spherical harmonics. By leveraging this property, we significantly expand the family of candidate quadrature rules and establish a connection between the geometrical information of the quadrature points and the error analysis of the method. We then utilize product integral rules to evaluate the singular integral, with the integrand being the product of the singular kernel and each spherical harmonic. We derive a practical [math] error bound, which consists of two terms: one controlled by the best approximation of the product of the continuous kernel and the solution and the other characterized by the Marcinkiewicz–Zygmund property and the best approximation polynomial of this product. Numerical examples validate our numerical analysis.
{"title":"Spherical Configurations and Quadrature Methods for Integral Equations of the Second Kind","authors":"Congpei An, Hao-Ning Wu","doi":"10.1137/24m1688370","DOIUrl":"https://doi.org/10.1137/24m1688370","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 148-169, February 2026. <br/> Abstract. In this paper, we propose and analyze a product integration method for the second-kind integral equation with weakly singular and continuous kernels on the unit sphere [math]. We employ quadrature rules that satisfy the Marcinkiewicz–Zygmund property to construct hyperinterpolation for approximating the product of the continuous kernel and the solution in terms of spherical harmonics. By leveraging this property, we significantly expand the family of candidate quadrature rules and establish a connection between the geometrical information of the quadrature points and the error analysis of the method. We then utilize product integral rules to evaluate the singular integral, with the integrand being the product of the singular kernel and each spherical harmonic. We derive a practical [math] error bound, which consists of two terms: one controlled by the best approximation of the product of the continuous kernel and the solution and the other characterized by the Marcinkiewicz–Zygmund property and the best approximation polynomial of this product. Numerical examples validate our numerical analysis.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"92 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101984","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 64, Issue 1, Page 125-147, February 2026. Abstract. Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the [math] projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing preasymptotic subgeometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.
{"title":"Scaling Optimized Hermite Approximation Methods","authors":"Hao Hu, Haijun Yu","doi":"10.1137/25m1737146","DOIUrl":"https://doi.org/10.1137/25m1737146","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 125-147, February 2026. <br/> Abstract. Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the [math] projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing preasymptotic subgeometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"286 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146089893","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 64, Issue 1, Page 76-102, February 2026. Abstract. The commonly used velocity discretization for simulating the radiative transport equation (RTE) is the discrete ordinates method (DOM). One of the long-standing drawbacks of DOM is the phenomenon known as the ray effect. Due to the high dimensionality of the RTE, DOM results in a large algebraic system to solve. The source iteration (SI) method is the most standard iterative method for solving this system. In this paper, by introducing randomness into the SI method, we propose a novel random source iteration (RSI) method that offers a new way to mitigate the ray effect without increasing the computational cost. We have rigorously proved that RSI is unbiased with respect to the SI method and that its variance is uniformly bounded across iteration steps; thus, the convergence order with respect to the number of samples is [math]. Furthermore, we prove that the RSI iteration process, as a Markov chain, is ergodic under mild assumptions. Numerical examples are presented to demonstrate the convergence of RSI and its effectiveness in mitigating the ray effect.
{"title":"Random Source Iteration Method: Mitigating the Ray Effect in the Discrete Ordinates Method","authors":"Jingyi Fu, Lei Li, Min Tang","doi":"10.1137/24m1669748","DOIUrl":"https://doi.org/10.1137/24m1669748","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 76-102, February 2026. <br/> Abstract. The commonly used velocity discretization for simulating the radiative transport equation (RTE) is the discrete ordinates method (DOM). One of the long-standing drawbacks of DOM is the phenomenon known as the ray effect. Due to the high dimensionality of the RTE, DOM results in a large algebraic system to solve. The source iteration (SI) method is the most standard iterative method for solving this system. In this paper, by introducing randomness into the SI method, we propose a novel random source iteration (RSI) method that offers a new way to mitigate the ray effect without increasing the computational cost. We have rigorously proved that RSI is unbiased with respect to the SI method and that its variance is uniformly bounded across iteration steps; thus, the convergence order with respect to the number of samples is [math]. Furthermore, we prove that the RSI iteration process, as a Markov chain, is ergodic under mild assumptions. Numerical examples are presented to demonstrate the convergence of RSI and its effectiveness in mitigating the ray effect.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070418","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 64, Issue 1, Page 103-124, February 2026. Abstract. We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.
{"title":"Second Order in Time Finite Element Schemes for Curve Shortening Flow and Curve Diffusion","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.1137/25m1737523","DOIUrl":"https://doi.org/10.1137/25m1737523","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 103-124, February 2026. <br/> Abstract. We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070417","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 64, Issue 1, Page 55-75, February 2026. Abstract. The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or magneto-quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities, and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level, and a full convergence analysis of the resulting finite element schemes is presented, indicating order-optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We also prove local quadratic convergence in a mesh size and polynomial degree–dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.
{"title":"On the Convergence of Higher-Order Finite Element Methods for Nonlinear Magnetostatics","authors":"H. Egger, F. Engertsberger, B. Radu","doi":"10.1137/24m168814x","DOIUrl":"https://doi.org/10.1137/24m168814x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 55-75, February 2026. <br/> Abstract. The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or magneto-quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities, and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level, and a full convergence analysis of the resulting finite element schemes is presented, indicating order-optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We also prove local quadratic convergence in a mesh size and polynomial degree–dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"24 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145995151","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 64, Issue 1, Page 29-54, February 2026. Abstract. We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite element discretizations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned matrix to be close to the identity and covers DD subdomains of arbitrary size, arbitrary absorbing layers/boundary conditions on both the global and local Helmholtz problems, and coarse spaces not necessarily related to the subdomains. The assumptions on the coarse space are satisfied by the approximation spaces using problem-adapted basis functions that have been recently analyzed as coarse spaces for the Helmholtz equation, as well as all spaces in which the Galerkin solutions are known to be quasi-optimal via a Schatz-type argument. As an example, we apply this theory when the coarse space consists of piecewise polynomials; these are then the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.
{"title":"Convergence Theory for Two-Level Hybrid Schwarz Preconditioners for High-Frequency Helmholtz Problems","authors":"J. Galkowski, E. A. Spence","doi":"10.1137/25m1726972","DOIUrl":"https://doi.org/10.1137/25m1726972","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 29-54, February 2026. <br/> Abstract. We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite element discretizations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned matrix to be close to the identity and covers DD subdomains of arbitrary size, arbitrary absorbing layers/boundary conditions on both the global and local Helmholtz problems, and coarse spaces not necessarily related to the subdomains. The assumptions on the coarse space are satisfied by the approximation spaces using problem-adapted basis functions that have been recently analyzed as coarse spaces for the Helmholtz equation, as well as all spaces in which the Galerkin solutions are known to be quasi-optimal via a Schatz-type argument. As an example, we apply this theory when the coarse space consists of piecewise polynomials; these are then the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145919768","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 64, Issue 1, Page 1-28, February 2026. Abstract. In this paper, we show that the approximation of high-dimensional functions, which are effectively low dimensional, does not suffer from the curse of dimensionality. This is shown first in a general reproducing kernel Hilbert space setup and then specifically for Sobolev and mixed-regularity Sobolev spaces. Finally, efficient estimates are derived for deciding whether a high-dimensional function is effectively low dimensional by studying error bounds in weighted reproducing kernel Hilbert spaces. The results are applied to parametric partial differential equations, a typical problem from uncertainty quantification.
{"title":"Constructive Approximation of High-Dimensional Functions with Small Efficient Dimension with Applications in Uncertainty Quantification","authors":"Christian Rieger, Holger Wendland","doi":"10.1137/24m171231x","DOIUrl":"https://doi.org/10.1137/24m171231x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 1-28, February 2026. <br/> Abstract. In this paper, we show that the approximation of high-dimensional functions, which are effectively low dimensional, does not suffer from the curse of dimensionality. This is shown first in a general reproducing kernel Hilbert space setup and then specifically for Sobolev and mixed-regularity Sobolev spaces. Finally, efficient estimates are derived for deciding whether a high-dimensional function is effectively low dimensional by studying error bounds in weighted reproducing kernel Hilbert spaces. The results are applied to parametric partial differential equations, a typical problem from uncertainty quantification.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145903754","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 2421-2453, December 2025. Abstract. We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments, and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine series (COS) method. The classical COS method is numerically very efficient in one dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a COS. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.
{"title":"From Characteristic Functions to Multivariate Distribution Functions and European Option Prices by the (Damped) COS Method","authors":"Gero Junike, Hauke Stier","doi":"10.1137/24m1666240","DOIUrl":"https://doi.org/10.1137/24m1666240","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2421-2453, December 2025. <br/> Abstract. We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments, and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine series (COS) method. The classical COS method is numerically very efficient in one dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a COS. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"157 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771639","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 2483-2511, December 2025. Abstract. The supremizer method enriches the reduced velocity basis for pressure recovery in incompressible flows, ensuring the inf-sup condition in the reduced space. In the full-order model, a small penalty term is often introduced to prevent spurious modes [Y. He, Math. Comp., 74 (2005), pp. 1201–1216] and is also essential for accuracy in the proper orthogonal decomposition–based reduced-order model [A.-L. Gerner and K. Veroy, Math. Models Methods Appl. Sci., 21 (2011), pp. 2103–2134]. However, coupling pressure and velocity, along with the supremizer basis, significantly increases the computational costs in both offline and online phases. We find that the primary role of supremizers is to improve stability, rather than velocity accuracy. We propose a novel method using several supremizers for the velocity basis, decoupling the penalized system to solve for velocity. The full set of supremizers is then used to recover pressure. This strategy reduces the computational cost while maintaining stability and accuracy. We derive error estimates using a supremizer-augmented projection operator, which depend on the inf-sup constant rather than on the inverse of the penalty coefficient. We also develop two new supremizer construction options satisfying the inf-sup condition, one of which avoids solving the full-order equations for obtaining supremizer basis, further reducing offline costs. Numerical experiments demonstrate the effectiveness of the proposed method. For comparable accuracy, CPU time tests show that the online computational cost is reduced by about [math], and the offline assembly cost by [math], compared to [Y. He, Math. Comp., 74 (2005), pp. 1201–1216].
SIAM数值分析杂志，第63卷，第6期，2483-2511页，2025年12月。摘要。超压器方法丰富了不可压缩流动中压力恢复的降速基础，保证了降速空间内的升压条件。在全阶模型中，通常引入一个小的惩罚项来防止伪模[Y]。他数学。Comp., 74 (2005), pp. 1201-1216]并且对于适当的基于正交分解的降阶模型的准确性也是必不可少的[A.-L.]Gerner和K. verroy，数学。模型、方法、应用。科学。， 21 (2011), pp. 2103-2134]。然而，耦合压力和速度，以及超喷器的基础，大大增加了离线和在线阶段的计算成本。我们发现，上位器的主要作用是提高稳定性，而不是速度精度。我们提出了一种新的方法，使用几个速度基的最优器，解耦惩罚系统来求解速度。然后使用全套的增压器来恢复压力。该策略在保持稳定性和准确性的同时降低了计算成本。我们使用超增广投影算子推导误差估计，它依赖于中-sup常数而不是惩罚系数的倒数。我们还开发了两种满足上料条件的新型上料器结构方案，其中一种方案避免了求解上料器基的全阶方程，进一步降低了离线成本。数值实验证明了该方法的有效性。对于类似的精度，CPU时间测试表明，与[Y]相比，在线计算成本减少了大约[math]，离线组装成本减少了[math]。他数学。《比较》，74 (2005),pp. 1201-1216。
{"title":"Fast Supremizer Method on Penalty-Based Reduced-Order Modeling for Incompressible Flows","authors":"Hui Yao, Mejdi Azaiez","doi":"10.1137/25m1746112","DOIUrl":"https://doi.org/10.1137/25m1746112","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2483-2511, December 2025. <br/> Abstract. The supremizer method enriches the reduced velocity basis for pressure recovery in incompressible flows, ensuring the inf-sup condition in the reduced space. In the full-order model, a small penalty term is often introduced to prevent spurious modes [Y. He, Math. Comp., 74 (2005), pp. 1201–1216] and is also essential for accuracy in the proper orthogonal decomposition–based reduced-order model [A.-L. Gerner and K. Veroy, Math. Models Methods Appl. Sci., 21 (2011), pp. 2103–2134]. However, coupling pressure and velocity, along with the supremizer basis, significantly increases the computational costs in both offline and online phases. We find that the primary role of supremizers is to improve stability, rather than velocity accuracy. We propose a novel method using several supremizers for the velocity basis, decoupling the penalized system to solve for velocity. The full set of supremizers is then used to recover pressure. This strategy reduces the computational cost while maintaining stability and accuracy. We derive error estimates using a supremizer-augmented projection operator, which depend on the inf-sup constant rather than on the inverse of the penalty coefficient. We also develop two new supremizer construction options satisfying the inf-sup condition, one of which avoids solving the full-order equations for obtaining supremizer basis, further reducing offline costs. Numerical experiments demonstrate the effectiveness of the proposed method. For comparable accuracy, CPU time tests show that the online computational cost is reduced by about [math], and the offline assembly cost by [math], compared to [Y. He, Math. Comp., 74 (2005), pp. 1201–1216].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771690","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 2454-2482, December 2025. Abstract. We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE, we introduce square-squeezing—a homeomorphic multilinear hypercube-simplex transformation—reparametrizing an initial flat triangulation of the manifold to a cubical mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparametrized mesh through interpolation in Chebyshev–Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the [math]-order total variation of the integrand and the surface parametrization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing [math], the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing [math]-refinements to overcome the limitations of [math]-refinements for highly varying smooth integrals.
{"title":"High-Order Integration on Regular Triangulated Manifolds Reaches Superalgebraic Approximation Rates Through Cubical Reparametrizations","authors":"Gentian Zavalani, Oliver Sander, Michael Hecht","doi":"10.1137/24m1707274","DOIUrl":"https://doi.org/10.1137/24m1707274","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 6, Page 2454-2482, December 2025. <br/> Abstract. We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE, we introduce square-squeezing—a homeomorphic multilinear hypercube-simplex transformation—reparametrizing an initial flat triangulation of the manifold to a cubical mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparametrized mesh through interpolation in Chebyshev–Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the [math]-order total variation of the integrand and the surface parametrization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing [math], the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing [math]-refinements to overcome the limitations of [math]-refinements for highly varying smooth integrals.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"111 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145771636","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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