SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 619-640, April 2025. Abstract. Mesh quality is crucial in the simulation of surface evolution equations using parametric finite element methods (FEMs). Energy-diminishing schemes may fail even when the surface remains smooth due to poor mesh distribution. In this paper, we aim to develop mesh-preserving and energy-stable parametric finite element schemes for the mean curvature flow and surface diffusion of two-dimensional surfaces. These new schemes are based on a reformulation of general surface evolution equations, achieved by coupling the original equation with a modified harmonic map heat flow. We demonstrate that our Euler schemes are energy-diminishing, and the proposed BDF2 schemes are energy-stable under a mild assumption on the mesh distortion. Numerical tests demonstrate that the proposed schemes perform exceptionally well in maintaining mesh quality.
{"title":"Mesh-Preserving and Energy-Stable Parametric FEM for Geometric Flows of Surfaces","authors":"Beiping Duan","doi":"10.1137/24m1671542","DOIUrl":"https://doi.org/10.1137/24m1671542","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 619-640, April 2025. <br/> Abstract. Mesh quality is crucial in the simulation of surface evolution equations using parametric finite element methods (FEMs). Energy-diminishing schemes may fail even when the surface remains smooth due to poor mesh distribution. In this paper, we aim to develop mesh-preserving and energy-stable parametric finite element schemes for the mean curvature flow and surface diffusion of two-dimensional surfaces. These new schemes are based on a reformulation of general surface evolution equations, achieved by coupling the original equation with a modified harmonic map heat flow. We demonstrate that our Euler schemes are energy-diminishing, and the proposed BDF2 schemes are energy-stable under a mild assumption on the mesh distortion. Numerical tests demonstrate that the proposed schemes perform exceptionally well in maintaining mesh quality.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"11 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143723877","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}
Rodolfo Araya, Christopher Harder, Abner H. Poza, Frédéric Valentin
SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 588-618, April 2025. Abstract. The multiscale hybrid-mixed (MHM) method for the Stokes operator was formally introduced in [R. Araya et al., Comput. Methods Appl. Mech. Engrg., 324, pp. 29–53, 2017] and numerically validated. The method has face degrees of freedom associated with multiscale basis functions computed from local Neumann problems driven by discontinuous polynomial spaces on skeletal meshes. The two-level MHM version approximates the multiscale basis using a stabilized finite element method. This work proposes the first numerical analysis for the one- and two-level MHM method applied to the Stokes/Brinkman equations within a new abstract framework relating MHM methods to discrete primal hybrid formulations. As a result, we generalize the two-level MHM method to include general second-level solvers and continuous polynomial interpolation on faces and establish abstract conditions to have those methods well-posed and optimally convergent on natural norms. We apply the abstract setting to analyze the MHM methods using stabilized and stable finite element methods as second-level solvers with (dis)continuous interpolation on faces. Also, we find that the discrete velocity and pressure variables preserve the balance of forces and conservation of mass at the element level. Numerical benchmarks assess theoretical results.
{"title":"Multiscale Hybrid-Mixed Methods for the Stokes and Brinkman Equations—A Priori Analysis","authors":"Rodolfo Araya, Christopher Harder, Abner H. Poza, Frédéric Valentin","doi":"10.1137/24m1649368","DOIUrl":"https://doi.org/10.1137/24m1649368","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 588-618, April 2025. <br/> Abstract. The multiscale hybrid-mixed (MHM) method for the Stokes operator was formally introduced in [R. Araya et al., Comput. Methods Appl. Mech. Engrg., 324, pp. 29–53, 2017] and numerically validated. The method has face degrees of freedom associated with multiscale basis functions computed from local Neumann problems driven by discontinuous polynomial spaces on skeletal meshes. The two-level MHM version approximates the multiscale basis using a stabilized finite element method. This work proposes the first numerical analysis for the one- and two-level MHM method applied to the Stokes/Brinkman equations within a new abstract framework relating MHM methods to discrete primal hybrid formulations. As a result, we generalize the two-level MHM method to include general second-level solvers and continuous polynomial interpolation on faces and establish abstract conditions to have those methods well-posed and optimally convergent on natural norms. We apply the abstract setting to analyze the MHM methods using stabilized and stable finite element methods as second-level solvers with (dis)continuous interpolation on faces. Also, we find that the discrete velocity and pressure variables preserve the balance of forces and conservation of mass at the element level. Numerical benchmarks assess theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143618495","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}
Kai Jiang, Xueyang Li, Yao Ma, Juan Zhang, Pingwen Zhang, Qi Zhou
SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 564-587, April 2025. Abstract. In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to one-dimensional, two-dimensional (2D), and three-dimensional (3D) quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically verified.
{"title":"Irrational-Window-Filter Projection Method and Application to Quasiperiodic Schrödinger Eigenproblems","authors":"Kai Jiang, Xueyang Li, Yao Ma, Juan Zhang, Pingwen Zhang, Qi Zhou","doi":"10.1137/24m1666197","DOIUrl":"https://doi.org/10.1137/24m1666197","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 564-587, April 2025. <br/> Abstract. In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to one-dimensional, two-dimensional (2D), and three-dimensional (3D) quasiperiodic Schrödinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically verified.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"86 1 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599756","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 2, Page 542-563, April 2025. Abstract. Efficient simulation of stochastic partial differential equations (SPDEs) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain. The Gaussian noise is white in time, spatially correlated, and modeled as a standard cylindrical Wiener process on a reproducing kernel Hilbert space. This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel. The kernel is assumed to be defined on a larger regular domain, allowing for sampling by the circulant embedding method. The error bound under mild kernel assumptions requires nontrivial techniques like Hilbert–Schmidt bounds on products of finite element interpolants, entropy numbers of fractional Sobolev space embeddings, and an error bound for interpolants in fractional Sobolev norms. Examples with kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Key findings include the following: noise interpolation does not introduce additional errors for Matérn kernels in [math]; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.
{"title":"Piecewise Linear Interpolation of Noise in Finite Element Approximations of Parabolic SPDEs","authors":"Gabriel J. Lord, Andreas Petersson","doi":"10.1137/23m1574117","DOIUrl":"https://doi.org/10.1137/23m1574117","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 542-563, April 2025. <br/> Abstract. Efficient simulation of stochastic partial differential equations (SPDEs) on general domains requires noise discretization. This paper employs piecewise linear interpolation of noise in a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polyhedral domain. The Gaussian noise is white in time, spatially correlated, and modeled as a standard cylindrical Wiener process on a reproducing kernel Hilbert space. This paper provides the first rigorous analysis of the resulting noise discretization error for a general spatial covariance kernel. The kernel is assumed to be defined on a larger regular domain, allowing for sampling by the circulant embedding method. The error bound under mild kernel assumptions requires nontrivial techniques like Hilbert–Schmidt bounds on products of finite element interpolants, entropy numbers of fractional Sobolev space embeddings, and an error bound for interpolants in fractional Sobolev norms. Examples with kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Key findings include the following: noise interpolation does not introduce additional errors for Matérn kernels in [math]; there exist kernels that yield dominant interpolation errors; and generating noise on a coarser mesh does not always compromise accuracy.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"68 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143582914","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}
Julian Braun, Christoph Ortner, Yangshuai Wang, Lei Zhang
SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 520-541, April 2025. Abstract. Crystalline materials exhibit long-range elastic fields due to the presence of defects, leading to significant domain size effects in atomistic simulations. A rigorous far-field expansion of these long-range fields identifies low-rank structure in the form of a sum of discrete multipole terms and continuum predictors [J. Braun, T. Hudson, and C. Ortner, Arch. Ration. Mech. Anal., 245 (2022), pp. 1437–1490]. We propose a novel numerical scheme that exploits this low-rank structure to accelerate material defect simulations by minimizing the domain size effects. Our approach iteratively improves the boundary condition, systematically following the asymptotic expansion of the far field. We provide both rigorous error estimates for the method and a range of empirical numerical tests to assess its convergence and robustness.
{"title":"Higher-Order Far-Field Boundary Conditions for Crystalline Defects","authors":"Julian Braun, Christoph Ortner, Yangshuai Wang, Lei Zhang","doi":"10.1137/24m165836x","DOIUrl":"https://doi.org/10.1137/24m165836x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 520-541, April 2025. <br/> Abstract. Crystalline materials exhibit long-range elastic fields due to the presence of defects, leading to significant domain size effects in atomistic simulations. A rigorous far-field expansion of these long-range fields identifies low-rank structure in the form of a sum of discrete multipole terms and continuum predictors [J. Braun, T. Hudson, and C. Ortner, Arch. Ration. Mech. Anal., 245 (2022), pp. 1437–1490]. We propose a novel numerical scheme that exploits this low-rank structure to accelerate material defect simulations by minimizing the domain size effects. Our approach iteratively improves the boundary condition, systematically following the asymptotic expansion of the far field. We provide both rigorous error estimates for the method and a range of empirical numerical tests to assess its convergence and robustness.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143570457","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 2, Page 495-519, April 2025. Abstract. Gaussian process regression is a classical kernel method for function estimation and data interpolation. In large data applications, computational costs can be reduced using low-rank or sparse approximations of the kernel. This paper investigates the effect of such kernel approximations on the interpolation error. We introduce a unified framework to analyze Gaussian process regression under important classes of computational misspecification: Karhunen–Loève expansions that result in low-rank kernel approximations, multiscale wavelet expansions that induce sparsity in the covariance matrix, and finite element representations that induce sparsity in the precision matrix. Our theory also accounts for epistemic misspecification in the choice of kernel parameters.
{"title":"Gaussian Process Regression under Computational and Epistemic Misspecification","authors":"Daniel Sanz-Alonso, Ruiyi Yang","doi":"10.1137/23m1624749","DOIUrl":"https://doi.org/10.1137/23m1624749","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 495-519, April 2025. <br/> Abstract. Gaussian process regression is a classical kernel method for function estimation and data interpolation. In large data applications, computational costs can be reduced using low-rank or sparse approximations of the kernel. This paper investigates the effect of such kernel approximations on the interpolation error. We introduce a unified framework to analyze Gaussian process regression under important classes of computational misspecification: Karhunen–Loève expansions that result in low-rank kernel approximations, multiscale wavelet expansions that induce sparsity in the covariance matrix, and finite element representations that induce sparsity in the precision matrix. Our theory also accounts for epistemic misspecification in the choice of kernel parameters.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"91 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143561189","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 2, Page 469-494, April 2025. Abstract. In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation over general regions in the complex plane using the monomial basis. Our analysis also yields a new upper bound for the condition number of an arbitrary Vandermonde matrix, which generalizes several previous results.
{"title":"On Polynomial Interpolation in the Monomial Basis","authors":"Zewen Shen, Kirill Serkh","doi":"10.1137/23m1623215","DOIUrl":"https://doi.org/10.1137/23m1623215","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 469-494, April 2025. <br/> Abstract. In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation over general regions in the complex plane using the monomial basis. Our analysis also yields a new upper bound for the condition number of an arbitrary Vandermonde matrix, which generalizes several previous results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546179","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 1, Page 437-460, February 2025. Abstract. We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart–Thomas functions, which is known from the literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [IMA J. Numer. Anal., 43 (2022), pp. 692–736] no longer holds. Even weaker [math]-convergence results no longer hold because the recovery sequences generally need to attain noninteger values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart–Thomas ansatz, which enables us to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.
SIAM数值分析杂志，第63卷，第1期，第437-460页，2025年2月。摘要。引入了具有全变分正则化和优化变量完整性约束的无限维优化问题的离散化方法。利用文献中已知的关于某些凸问题的Raviart-Thomas函数，提出了全变分项的对偶形式的离散化。由于我们有一个完整性约束，以前的分析由Caillaud和Chambolle [IMA J. number]。分析的。， 43 (2022), pp. 692-736]不再成立。甚至较弱的[数学]收敛结果也不再成立，因为恢复序列通常需要获得非整数值才能恢复极限函数的总变化。我们通过在嵌入的更细的网格上引入输入函数的离散化来解决这个问题。网格尺寸的超线性耦合意味着对Raviart-Thomas ansatz的粗网格进行平均，这使我们能够恢复整值极限函数与整值离散输入函数的总变化。此外，我们还可以通过恢复序列的极限总变分和依赖于网格尺寸比的误差来估计恢复序列的离散化总变分。对于离散优化问题，我们额外增加了一个约束，该约束在极限中消失，并强制最小化序列的紧性，从而使它们收敛到原始问题的最小化。这个约束包含一个可接受范围已确定的自由度。它的选择可能对实践中的解决方案有很大的影响，正如我们用成像的例子所展示的那样。
{"title":"Discretization of Total Variation in Optimization with Integrality Constraints","authors":"Annika Schiemann, Paul Manns","doi":"10.1137/24m164608x","DOIUrl":"https://doi.org/10.1137/24m164608x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 437-460, February 2025. <br/> Abstract. We introduce discretizations of infinite-dimensional optimization problems with total variation regularization and integrality constraints on the optimization variables. We advance the discretization of the dual formulation of the total variation term with Raviart–Thomas functions, which is known from the literature for certain convex problems. Since we have an integrality constraint, the previous analysis from Caillaud and Chambolle [IMA J. Numer. Anal., 43 (2022), pp. 692–736] no longer holds. Even weaker [math]-convergence results no longer hold because the recovery sequences generally need to attain noninteger values to recover the total variation of the limit function. We solve this issue by introducing a discretization of the input functions on an embedded, finer mesh. A superlinear coupling of the mesh sizes implies an averaging on the coarser mesh of the Raviart–Thomas ansatz, which enables us to recover the total variation of integer-valued limit functions with integer-valued discretized input functions. Moreover, we are able to estimate the discretized total variation of the recovery sequence by the total variation of its limit and an error depending on the mesh size ratio. For the discretized optimization problems, we additionally add a constraint that vanishes in the limit and enforces compactness of the sequence of minimizers, which yields their convergence to a minimizer of the original problem. This constraint contains a degree of freedom whose admissible range is determined. Its choice may have a strong impact on the solutions in practice as we demonstrate with an example from imaging.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143495222","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 1, Page 461-467, February 2025. Abstract. Various nonlinear Schwarz domain decomposition methods were proposed to solve the one-dimensional equidistribution principle in [M. J. Gander and R. D. Haynes, SIAM J. Numer. Anal., 50 (2012), pp. 2111-2135]. A corrected proof of convergence for the linearized Schwarz algorithm presented in section 3.2, under additional hypotheses, is presented here. An alternative linearized Schwarz algorithm for equidistributed grid generation is also provided.
SIAM数值分析杂志，第63卷，第1期，第461-467页，2025年2月。摘要。提出了多种非线性Schwarz域分解方法来求解[M]中的一维均匀分布原理。J.甘德和R. D.海恩斯，SIAM J. number。分析的。， 50 (2012), pp. 2111-2135]。第3.2节中给出的线性化Schwarz算法在附加假设下的收敛性的修正证明，在这里给出。本文还提供了一种可选的用于等分布网格生成的线性化Schwarz算法。
{"title":"Corrigendum: Domain Decomposition Approaches for Mesh Generation via the Equidistribution Principle","authors":"Martin J. Gander, Ronald D. Haynes, Felix Kwok","doi":"10.1137/24m1693453","DOIUrl":"https://doi.org/10.1137/24m1693453","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 461-467, February 2025. <br/> Abstract. Various nonlinear Schwarz domain decomposition methods were proposed to solve the one-dimensional equidistribution principle in [M. J. Gander and R. D. Haynes, SIAM J. Numer. Anal., 50 (2012), pp. 2111-2135]. A corrected proof of convergence for the linearized Schwarz algorithm presented in section 3.2, under additional hypotheses, is presented here. An alternative linearized Schwarz algorithm for equidistributed grid generation is also provided.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"83 1 Pt 2 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143495221","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 1, Page 422-436, February 2025. Abstract. Starting from an A-stable rational approximation to [math] of order [math], [math], families of stable methods are proposed to time discretize abstract IVPs of the type [math]. These numerical procedures turn out to be of order [math], thus overcoming the order reduction phenomenon, and only one evaluation of [math] per step is required.
{"title":"Rational Methods for Abstract, Linear, Nonhomogeneous Problems without Order Reduction","authors":"Carlos Arranz-Simón, César Palencia","doi":"10.1137/24m165942x","DOIUrl":"https://doi.org/10.1137/24m165942x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 1, Page 422-436, February 2025. <br/> Abstract. Starting from an A-stable rational approximation to [math] of order [math], [math], families of stable methods are proposed to time discretize abstract IVPs of the type [math]. These numerical procedures turn out to be of order [math], thus overcoming the order reduction phenomenon, and only one evaluation of [math] per step is required.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143485943","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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