IMA Journal of Numerical Analysis最新文献

Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions 尼斯特伦离散化亥姆霍兹分解边界积分方程公式的稳定性估计，用于解决二维纳维散射问题（带德里赫特边界条件）

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-11-10 DOI: 10.1093/imanum/drae078

Víctor Domínguez, Catalin Turc

{"title":"Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions","authors":"Víctor Domínguez, Catalin Turc","doi":"10.1093/imanum/drae078","DOIUrl":"https://doi.org/10.1093/imanum/drae078","url":null,"abstract":"Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142597484","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

Positive definite functions on a regular domain 规则域上的正定函数

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-11-06 DOI: 10.1093/imanum/drae074

Martin Buhmann, Yuan Xu

引用次数: 0

An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation 将近似分量模式合成法扩展至异质亥姆霍兹方程

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-10-26 DOI: 10.1093/imanum/drae076

Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom

{"title":"An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation","authors":"Elena Giammatteo, Alexander Heinlein, Matthias Schlottbom","doi":"10.1093/imanum/drae076","DOIUrl":"https://doi.org/10.1093/imanum/drae076","url":null,"abstract":"In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142490355","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

Time-dependent electromagnetic scattering from dispersive materials 色散材料随时间变化的电磁散射

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-10-25 DOI: 10.1093/imanum/drae071

Jörg Nick, Selina Burkhard, Christian Lubich

引用次数: 0

An exponential stochastic Runge–Kutta type method of order up to 1.5 for SPDEs of Nemytskii-type 针对 Nemytski 型 SPDE 的阶次高达 1.5 的指数随机 Runge-Kutta 类型方法

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-10-17 DOI: 10.1093/imanum/drae064

Claudine von Hallern, Ricarda Missfeldt, Andreas Rössler

引用次数: 0

Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and approximation properties 三维对流亥姆霍兹方程的三种准特雷弗茨函数：构造和近似特性

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-10-10 DOI: 10.1093/imanum/drae060

Lise-Marie Imbert-Gérard, Guillaume Sylvand

引用次数: 0

A convergent stochastic scalar auxiliary variable method 收敛的随机标量辅助变量法

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-10-10 DOI: 10.1093/imanum/drae065

Stefan Metzger

{"title":"A convergent stochastic scalar auxiliary variable method","authors":"Stefan Metzger","doi":"10.1093/imanum/drae065","DOIUrl":"https://doi.org/10.1093/imanum/drae065","url":null,"abstract":"We discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. (2018, The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys., 353, 407–416) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable this approach allows to derive a linear scheme while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen–Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our augmented scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time we are able to prove convergence of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski’s generalization of Skorokhod’s theorem. A generalization of the Gyöngy–Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen–Cahn equation. Finally, we present numerical simulations underlining the practicality of the scheme and the importance of the introduced augmentation terms.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142405030","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 priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow 三场准牛顿斯托克斯流混合 DG 方法的先验和后验误差分析

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-10-03 DOI: 10.1093/imanum/drae067

Lina Zhao

{"title":"A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow","authors":"Lina Zhao","doi":"10.1093/imanum/drae067","DOIUrl":"https://doi.org/10.1093/imanum/drae067","url":null,"abstract":"In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142369108","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 maximum-minimum principle for a linearly implicit scheme for nonlinear parabolic FEM problems under weakened time restrictions 弱化时间限制下非线性抛物线有限元问题线性隐式方案的离散最大最小原则

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-09-28 DOI: 10.1093/imanum/drae072

István Faragó, Róbert Horváth, János Karátson

{"title":"Discrete maximum-minimum principle for a linearly implicit scheme for nonlinear parabolic FEM problems under weakened time restrictions","authors":"István Faragó, Róbert Horváth, János Karátson","doi":"10.1093/imanum/drae072","DOIUrl":"https://doi.org/10.1093/imanum/drae072","url":null,"abstract":"In this paper, we extend our earlier results in Faragó, I., Karátson, J. and Korotov, S. (2012, Discrete maximum principles for nonlinear parabolic PDE systems. IMA J. Numer. Anal., 32, 1541–1573) on the discrete maximum-minimum principle (DMP) for nonlinear parabolic systems of PDEs. We propose a linearly implicit scheme, where only linear problems have to be solved on the time layers. We obtain a DMP without the restrictive condition $varDelta tle O(h^{2})$. We show that we only need the lower bound $varDelta tge O(h^{2})$, further, depending on the Lipschitz condition of the given nonlinearity, the upper bound is just $varDelta tle C$ (for globally Lipschitz) or $varDelta tle O(h^{gamma })$ (for locally Lipschitz) for some constant $C&gt;0$ arising from the PDE, or some $gamma &lt; 2$, respectively. In most situations in practical models, the latter condition becomes $varDelta t le O( h^{2/3} )$ in 2D and $varDelta t le O( h )$ in 3D. Various real-life examples are also presented where the results can be applied to obtain physically relevant numerical solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328968","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 unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach 基于广义 SAV 方法的卡恩-希利亚德方程的无条件耗能自适应 IMEX BDF2 方案及其误差估算

IF 2.1 2区数学

IMA Journal of Numerical Analysis Pub Date : 2024-09-25 DOI: 10.1093/imanum/drae057

Yifan Wei, Jiwei Zhang, Chengchao Zhao, Yanmin Zhao

{"title":"An unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn–Hilliard equation on generalized SAV approach","authors":"Yifan Wei, Jiwei Zhang, Chengchao Zhao, Yanmin Zhao","doi":"10.1093/imanum/drae057","DOIUrl":"https://doi.org/10.1093/imanum/drae057","url":null,"abstract":"An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn–Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., A1: $0&lt;r_{k}:=tau _{k}/tau _{k-1}&lt; r_{max }approx 4.8645$, we establish a rigorous error estimate in $H^{1}$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^{1}$-norm. The $H^{1}$ bound of numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142321471","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