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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
Convergence Analysis of the Monte Carlo Method for the Random Navier–Stokes–Fourier System 随机Navier-Stokes-Fourier系统蒙特卡罗方法的收敛性分析
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-06-18 DOI: 10.1137/23m1563232
Mária Lukáčová-Medviďová, Bangwei She, Yuhuan Yuan
{"title":"Convergence Analysis of the Monte Carlo Method for the Random Navier–Stokes–Fourier System","authors":"Mária Lukáčová-Medviďová, Bangwei She, Yuhuan Yuan","doi":"10.1137/23m1563232","DOIUrl":"https://doi.org/10.1137/23m1563232","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1254-1280, June 2025. <br/> Abstract. In the present paper, we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier–Stokes–Fourier system. The Monte Carlo method, frequently used for approximating statistical moments, is combined with a suitable deterministic discretization method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to the statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"38 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144311938","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 Approximation of Stationary Fokker–Planck–Kolmogorov Equations with Application to Periodic Numerical Homogenization 平稳Fokker-Planck-Kolmogorov方程的有限元逼近及其在周期数值均匀化中的应用
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-06-18 DOI: 10.1137/24m1692848
Timo Sprekeler, Endre Süli, Zhiwen Zhang
{"title":"Finite Element Approximation of Stationary Fokker–Planck–Kolmogorov Equations with Application to Periodic Numerical Homogenization","authors":"Timo Sprekeler, Endre Süli, Zhiwen Zhang","doi":"10.1137/24m1692848","DOIUrl":"https://doi.org/10.1137/24m1692848","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1315-1343, June 2025. <br/> Abstract. We propose and rigorously analyze a finite element method for the approximation of stationary Fokker–Planck–Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. In particular, the Cordes setting guarantees the existence and uniqueness of a square-integrable invariant measure. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings based on the finite element scheme for stationary FPK problems developed in the first part. Finally, we demonstrate the performance of the methods through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"100 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144311937","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
High-Order Sparse-PIC Methods: Analysis and Numerical Investigations 高阶稀疏pic方法：分析与数值研究
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-06-18 DOI: 10.1137/24m1665143
Fabrice Deluzet, Clément Guillet, Jacek Narski, Paul Pace
{"title":"High-Order Sparse-PIC Methods: Analysis and Numerical Investigations","authors":"Fabrice Deluzet, Clément Guillet, Jacek Narski, Paul Pace","doi":"10.1137/24m1665143","DOIUrl":"https://doi.org/10.1137/24m1665143","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1281-1314, June 2025. <br/> Abstract. Particle-in-cell (PIC) methods embedding sparse grids have been recently introduced to decrease the statistical noise inherent to PIC approximations. In sparse-PIC methods, the numerical noise is filtered out from the approximation thanks to a reconstruction of the grid quantities on a hierarchy of coarse meshes. This procedure introduces a significant gain in the precision of the numerical approximation with respect to the mean number of particles in a grid cell, this parameter controlling the numerical noise, but also introduces a slight discrepancy of the method precision with respect to the mesh resolution. In previous studies, this issue is addressed by a careful tuning of the grids composing the sparse grid hierarchy, to define a trade-off between the gain in the numerical noise and the loss in the grid error. The present work is dedicated to improving the precision of sparse-PIC methods with respect to the mesh resolution and, contrary to the previous achievements, without deteriorating the gains with respect to the statistical noise. A refined error estimate is proposed. It permits one to control the number of numerical particles to obtain a comparable statistical noise in the approximation carried out by either a standard or a sparse-PIC method (and thus assess the true merits of the methods).","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"600 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144319836","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 Analysis of a Solver for the Linear Poisson–Boltzmann Model 线性泊松-玻尔兹曼模型求解器的收敛性分析
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2025-06-10 DOI: 10.1137/24m1717087
Xuanyu Liu, Yvon Maday, Chaoyu Quan, Hui Zhang
{"title":"Convergence Analysis of a Solver for the Linear Poisson–Boltzmann Model","authors":"Xuanyu Liu, Yvon Maday, Chaoyu Quan, Hui Zhang","doi":"10.1137/24m1717087","DOIUrl":"https://doi.org/10.1137/24m1717087","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1232-1253, June 2025. <br/> Abstract. This work investigates the convergence of a domain decomposition method for the Poisson–Boltzmann model that can be formulated as an interior-exterior transmission problem. To study its convergence, we introduce an interior-exterior constant providing an upper bound of the [math] norm of any harmonic function in the interior, and establish a spectral equivalence for related Dirichlet-to-Neumann operators to estimate the spectrum of interior-exterior iteration operator. This analysis is nontrivial due to the unboundedness of the exterior subdomain, which distinguishes it from the classical analysis of the Schwarz alternating method with nonoverlapping bounded subdomains. It is proved that for the linear Poisson–Boltzmann solvent model in reality, the convergence of interior-exterior iteration is ensured when the relaxation parameter lies between 0 and 2. This convergence result interprets the good performance of ddLPB method developed in [C. Quan, B. Stamm, and Y. Maday, SIAM J. Sci. Comput., 41 (2019), pp. B320–B350] where the relaxation parameter is set to 1. Numerical simulations are conducted to verify our convergence analysis and to investigate the optimal relaxation parameter for the interior-exterior iteration.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"140 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144260242","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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