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Accurately Recover Global Quasiperiodic Systems by Finite Points 用有限点精确恢复全局准周期系统
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-24 DOI: 10.1137/23m1620247
Kai Jiang, Qi Zhou, Pingwen Zhang
{"title":"Accurately Recover Global Quasiperiodic Systems by Finite Points","authors":"Kai Jiang, Qi Zhou, Pingwen Zhang","doi":"10.1137/23m1620247","DOIUrl":"https://doi.org/10.1137/23m1620247","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024. <br/> Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higher-dimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141764227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Duality-Based Error Control for the Signorini Problem 基于对偶性的西格诺里尼问题误差控制
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-23 DOI: 10.1137/22m1534791
Ben S. Ashby, Tristan Pryer
{"title":"Duality-Based Error Control for the Signorini Problem","authors":"Ben S. Ashby, Tristan Pryer","doi":"10.1137/22m1534791","DOIUrl":"https://doi.org/10.1137/22m1534791","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1687-1712, August 2024. <br/> Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of the discretization error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in [math] for any [math]. We summarize extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions 具有非均质 Dirichlet 边界条件的稳定广义 Navier-Stokes 方程的有限元离散化
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-23 DOI: 10.1137/23m1607398
Julius Jeßberger, Alex Kaltenbach
{"title":"Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions","authors":"Julius Jeßberger, Alex Kaltenbach","doi":"10.1137/23m1607398","DOIUrl":"https://doi.org/10.1137/23m1607398","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1660-1686, August 2024. <br/> Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor 无对数因子的非连续伽勒金时间步进方法的离散最大正则性
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-22 DOI: 10.1137/23m1580802
Takahito Kashiwabara, Tomoya Kemmochi
{"title":"Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor","authors":"Takahito Kashiwabara, Tomoya Kemmochi","doi":"10.1137/23m1580802","DOIUrl":"https://doi.org/10.1137/23m1580802","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1638-1659, August 2024. <br/> Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a New Class of BDF and IMEX Schemes for Parabolic Type Equations 关于抛物型方程的一类新的 BDF 和 IMEX 方案
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-16 DOI: 10.1137/23m1612986
Fukeng Huang, Jie Shen
{"title":"On a New Class of BDF and IMEX Schemes for Parabolic Type Equations","authors":"Fukeng Huang, Jie Shen","doi":"10.1137/23m1612986","DOIUrl":"https://doi.org/10.1137/23m1612986","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1609-1637, August 2024. <br/> Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit schemes for parabolic type equations based on the Taylor expansions at time [math] with [math] being a tunable parameter. These new schemes, with a suitable [math], allow larger time steps at higher order for stiff problems than that which is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141631349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Localized Implicit Time Stepping for the Wave Equation 波方程的局部隐式时间步进
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-15 DOI: 10.1137/23m1582618
Dietmar Gallistl, Roland Maier
{"title":"Localized Implicit Time Stepping for the Wave Equation","authors":"Dietmar Gallistl, Roland Maier","doi":"10.1137/23m1582618","DOIUrl":"https://doi.org/10.1137/23m1582618","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1589-1608, August 2024. <br/> Abstract. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a restart is introduced after a certain number of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"47 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141625086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Full-Spectrum Dispersion Relation Preserving Summation-by-Parts Operators 全谱色散关系保全逐部求和算子
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-11 DOI: 10.1137/23m1586471
Christopher Williams, Kenneth Duru
{"title":"Full-Spectrum Dispersion Relation Preserving Summation-by-Parts Operators","authors":"Christopher Williams, Kenneth Duru","doi":"10.1137/23m1586471","DOIUrl":"https://doi.org/10.1137/23m1586471","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1565-1588, August 2024. <br/> Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of interior [math]-dispersion-relation-preserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete methodology to construct them. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which satisfy the summation-by-parts principle and preserve the dispersion relation of the continuous problem uniformly to an [math] error tolerance for their interior stencil. We give a general framework to design provably stable finite-difference operators whose interior stencil preserves the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ([math]-mode) present on any equidistant grid at a tolerance of [math] maximum error within the interior stencil, with minimal extra stencil points. As standard finite-difference schemes have a [math] dispersion error for high-frequency components, fine meshes must be used to resolve these components. Our derived schemes may compute solutions with the same accuracy as traditional schemes on far coarser meshes, which in high dimensions significantly improves the computational cost.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"6 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces 一般空间中最小过采样和插值的随机最小二乘法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-09 DOI: 10.1137/23m160178x
Matthieu Dolbeault, Moulay Abdellah Chkifa
{"title":"Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces","authors":"Matthieu Dolbeault, Moulay Abdellah Chkifa","doi":"10.1137/23m160178x","DOIUrl":"https://doi.org/10.1137/23m160178x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1515-1538, August 2024. <br/> Abstract. In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math] is larger than the dimension [math] of the approximation space by a constant multiplicative ratio. On the other hand, for [math], we obtain an interpolation strategy with a stability factor of order [math]. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009, pp. 255–262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315–2336], with polynomial computational complexity.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Robust Finite Elements for Linearized Magnetohydrodynamics 线性化磁流体力学的稳健有限元
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-09 DOI: 10.1137/23m1582783
L. Beirão da Veiga, F. Dassi, G. Vacca
{"title":"Robust Finite Elements for Linearized Magnetohydrodynamics","authors":"L. Beirão da Veiga, F. Dassi, G. Vacca","doi":"10.1137/23m1582783","DOIUrl":"https://doi.org/10.1137/23m1582783","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1539-1564, August 2024. <br/> Abstract. We introduce a pressure robust finite element method for the linearized magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a nonconforming BDM approach with suitable DG terms for the fluid part, combined with an [math]-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems 凸最小化问题混合高阶方法的离散弱对偶性
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-04 DOI: 10.1137/23m1594534
Ngoc Tien Tran
{"title":"Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems","authors":"Ngoc Tien Tran","doi":"10.1137/23m1594534","DOIUrl":"https://doi.org/10.1137/23m1594534","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1492-1514, August 2024. <br/> Abstract. This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a posteriori error estimates on regular triangulations into simplices using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs better compared to uniform mesh refinements.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"53 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141545843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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