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Error Analysis of BDF 1–6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates BDF - 6时间步进方法在瞬态斯托克斯问题中的误差分析：速度和压力估计
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-24 DOI: 10.1137/23m1606800
Alessandro Contri, Balázs Kovács, André Massing
{"title":"Error Analysis of BDF 1–6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates","authors":"Alessandro Contri, Balázs Kovács, André Massing","doi":"10.1137/23m1606800","DOIUrl":"https://doi.org/10.1137/23m1606800","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1586-1616, August 2025. <br/> Abstract. We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [SIAM J. Numer. Anal., 47 (2009), pp. 409–439] and provide a unified theoretical analysis of backward difference formula methods of orders 1 to 6. The main novelty of our approach lies in deriving optimal-order stability and error estimates for both the velocity and the pressure using Dahlquist’s [math]-stability concept together with the multiplier technique introduced by Nevanlinna and Odeh and recently by Akrivis et al. [SIAM J. Numer. Anal., 59 (2021), pp. 2449–2472]. When combined with a method-dependent Ritz projection for the initial data, unconditional stability can be shown, while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"119 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702055","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
Trefftz Discontinuous Galerkin Approximation of an Acoustic Waveguide 声波导的Trefftz不连续伽辽金近似
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-21 DOI: 10.1137/24m1686905
Peter Monk, Manuel Pena, Virginia Selgas
{"title":"Trefftz Discontinuous Galerkin Approximation of an Acoustic Waveguide","authors":"Peter Monk, Manuel Pena, Virginia Selgas","doi":"10.1137/24m1686905","DOIUrl":"https://doi.org/10.1137/24m1686905","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1561-1585, August 2025. <br/> Abstract. We propose a modified Trefftz discontinuous Galerkin (TDG) method for approximating a time-harmonic acoustic scattering problem in an infinitely elongated waveguide. In the waveguide we suppose that there is a bounded, penetrable, and possibly absorbing scatterer. The classical TDG is not applicable when the scatterer is absorbing. Novel features of our modified TDG method are that it is applicable in this case, and it uses a stable treatment of the outgoing radiation condition for the scattered field. For the modified TDG, we prove [math] and [math]-convergence in the [math] norm for nonabsorbing scatterers. The theoretical results are verified numerically for a discretization based on plane waves, and also investigated numerically for absorbing scatterers (in which case the plane waves are evanescent in the scatterer).","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144669721","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
A Posteriori Error Control for the Allen–Cahn Equation with Variable Mobility 变迁移率Allen-Cahn方程的后验误差控制
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-17 DOI: 10.1137/24m1646406
A. Brunk, J. Giesselmann, M. Lukáčová-Medvi[math]ová
{"title":"A Posteriori Error Control for the Allen–Cahn Equation with Variable Mobility","authors":"A. Brunk, J. Giesselmann, M. Lukáčová-Medvi[math]ová","doi":"10.1137/24m1646406","DOIUrl":"https://doi.org/10.1137/24m1646406","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1540-1560, August 2025. <br/> Abstract. In this work, we derive a [math]-robust a posteriori error estimator for finite element approximations of the Allen–Cahn equation with variable nondegenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144645433","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
Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion 拟线性次扩散的正则性分析及高阶时间步进格式
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-17 DOI: 10.1137/23m159531x
Bangti Jin, Qimeng Quan, Barbara Wohlmuth, Zhi Zhou
{"title":"Regularity Analysis and High-Order Time Stepping Scheme for Quasilinear Subdiffusion","authors":"Bangti Jin, Qimeng Quan, Barbara Wohlmuth, Zhi Zhou","doi":"10.1137/23m159531x","DOIUrl":"https://doi.org/10.1137/23m159531x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1512-1539, August 2025. <br/> Abstract. In this work, we investigate a quasilinear subdiffusion model which involves a fractional derivative of order [math] in time and a nonlinear diffusion coefficient. First, using smoothing properties of solution operators for linear subdiffusion and a perturbation argument, we prove several new pointwise-in-time Sobolev regularity estimates that are useful for numerical analysis. Then we develop a time-stepping scheme to solve quasilinear subdiffusion, based on convolution quadrature generated by the second-order backward differentiation formula with a correction at the first step. Further, we establish that the convergence order of the scheme is [math] without imposing any additional assumption on the regularity of the solution, which is high-order in the sense that its convergence rate is higher than the first-order convergence of the vanilla scheme. The analysis relies on refined Sobolev regularity of the nonlinear perturbation remainder and smoothing properties of discrete solution operators. Several numerical experiments in two space dimensions are presented to show the sharpness of the error estimate.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"84 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144645495","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
Convolution Quadrature for the Quasilinear Subdiffusion Equation 拟线性次扩散方程的卷积正交
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-15 DOI: 10.1137/23m161450x
Maria López-Fernández, Łukasz Płociniczak
{"title":"Convolution Quadrature for the Quasilinear Subdiffusion Equation","authors":"Maria López-Fernández, Łukasz Płociniczak","doi":"10.1137/23m161450x","DOIUrl":"https://doi.org/10.1137/23m161450x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1482-1511, August 2025. <br/> Abstract. We construct a convolution quadrature (CQ) scheme for the quasilinear subdiffusion equation of order [math] and supply it with the fast and oblivious implementation. In particular, we find a condition for the CQ to be admissible and discretize the spatial part of the equation with the finite element method. We prove the unconditional stability and convergence of the scheme and find a bound on the error. Our estimates are globally optimal for all [math] and pointwise for [math] in the sense that they reduce to the well-known results for the linear equation. For the semilinear case, our estimates are optimal both globally and locally. As a passing result, we also obtain a discrete Grönwall inequality for the CQ, which is a crucial ingredient in our convergence proof based on the energy method. The paper concludes with numerical examples verifying convergence and computation time reduction when using fast and oblivious quadrature.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144630011","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
Dynamic Ritz Projection of Mean Curvature Flow and Optimal [math] Convergence of Parametric FEM 平均曲率流的动态Ritz投影与参数有限元的最优收敛
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-14 DOI: 10.1137/24m1689053
Buyang Li, Rong Tang
{"title":"Dynamic Ritz Projection of Mean Curvature Flow and Optimal [math] Convergence of Parametric FEM","authors":"Buyang Li, Rong Tang","doi":"10.1137/24m1689053","DOIUrl":"https://doi.org/10.1137/24m1689053","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1454-1481, August 2025. <br/> Abstract. A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the [math] and [math] norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the [math] norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"101 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144629958","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
An Accurate and Efficient Scheme for Function Extension on Smooth Domains 光滑域上函数扩展的一种精确有效的方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-10 DOI: 10.1137/23m1622064
Charles L. Epstein, Fredrik Fryklund, Shidong Jiang
{"title":"An Accurate and Efficient Scheme for Function Extension on Smooth Domains","authors":"Charles L. Epstein, Fredrik Fryklund, Shidong Jiang","doi":"10.1137/23m1622064","DOIUrl":"https://doi.org/10.1137/23m1622064","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1427-1453, August 2025. <br/> Abstract. A new scheme is proposed to construct an [math]-times differentiable function extension of an [math]-times differentiable function defined on a smooth domain, [math] in [math]-dimensions. The extension scheme relies on an explicit formula consisting of a linear combination of [math] function values in [math] which extends the function along directions normal to the boundary. Smoothness tangent to the boundary is automatic. The performance of the scheme is illustrated by using function extension as part of a numerical solver for the Poisson equation on domains with complex geometry in both two and three dimensions. Although the cost of extending the function increases mildly with the extension order, it remains a small fraction of the overall algorithm. Moreover, the modest additional work required for high order function extensions leads to considerably more accurate solutions of the partial differential equation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144603408","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
Explicit Runge–Kutta Methods that Alleviate Order Reduction 显式龙格-库塔方法减轻序降
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-08 DOI: 10.1137/23m1606812
Abhijit Biswas, David I. Ketcheson, Steven Roberts, Benjamin Seibold, David Shirokoff
{"title":"Explicit Runge–Kutta Methods that Alleviate Order Reduction","authors":"Abhijit Biswas, David I. Ketcheson, Steven Roberts, Benjamin Seibold, David Shirokoff","doi":"10.1137/23m1606812","DOIUrl":"https://doi.org/10.1137/23m1606812","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1398-1426, August 2025. <br/> Abstract. Explicit Runge–Kutta (RK) methods are susceptible to a reduction in the observed order of convergence when applied to an initial boundary value problem with time-dependent boundary conditions. We study conditions on explicit RK methods that guarantee high order convergence for linear problems; we refer to these conditions as weak stage order conditions. We prove a general relationship between the method’s order, weak stage order, and number of stages. We derive explicit RK methods with high weak stage order and demonstrate, through numerical tests, that they avoid the order reduction phenomenon up to any order for linear problems and up to order three for nonlinear problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144586772","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
Convergence Estimates of Regularized Solutions to Inverse Space-Dependent Source Problems with Time-Dependent Boundary Measurement 具有时变边界测量的逆空间依赖源问题正则解的收敛性估计
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-07-04 DOI: 10.1137/24m1692885
Chunlong Sun, Wenlong Zhang
{"title":"Convergence Estimates of Regularized Solutions to Inverse Space-Dependent Source Problems with Time-Dependent Boundary Measurement","authors":"Chunlong Sun, Wenlong Zhang","doi":"10.1137/24m1692885","DOIUrl":"https://doi.org/10.1137/24m1692885","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1369-1397, August 2025. <br/> Abstract. In this work, we investigate the Tikhonov-type regularized solutions and their finite element solutions to the inverse space-dependent source problem from boundary measurement data. First, with the classical source condition, we establish the convergence of regularized solutions and their finite element solutions under the standard [math] norm. The error estimates present explicit dependence on the critical parameters like noise level, regularization parameter, mesh size, and time step size. Next, based on a proposed weak norm, we get the stability of Lipschitz type for the inverse problem, and then the first order convergence of regularized solutions can be derived in the sense of weak norm. We get this convergence without any source condition. Moreover, this work is carried out for the discrete data. We suppose that the observation points are discrete and the pointwise measurement data come with independent sub-Gaussian random noises. Then we give the stochastic convergence of regularized solutions and propose an efficient iterative algorithm to determine the optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithms.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"8 1","pages":"1369-1397"},"PeriodicalIF":2.9,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144566011","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
Ultra-Weak Least Squares Discretizations for Unique Continuation and Cauchy Problems 唯一延拓与柯西问题的超弱最小二乘离散
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-06-23 DOI: 10.1137/24m1674844
Harald Monsuur, Rob Stevenson
{"title":"Ultra-Weak Least Squares Discretizations for Unique Continuation and Cauchy Problems","authors":"Harald Monsuur, Rob Stevenson","doi":"10.1137/24m1674844","DOIUrl":"https://doi.org/10.1137/24m1674844","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1344-1368, June 2025. <br/> Abstract. In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial and test spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the [math]-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of [math]-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound or, in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144341262","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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