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IMA Journal of Numerical Analysis最新文献

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Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media 断裂多孔介质输运问题的整体和局部时间步进解耦算法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-04-03 DOI: 10.1093/imanum/drae005
Yanzhao Cao, Thi-Thao-Phuong Hoang, Phuoc-Toan Huynh
{"title":"Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media","authors":"Yanzhao Cao, Thi-Thao-Phuong Hoang, Phuoc-Toan Huynh","doi":"10.1093/imanum/drae005","DOIUrl":"https://doi.org/10.1093/imanum/drae005","url":null,"abstract":"The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140349099","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 the necessity of the inf-sup condition for a mixed finite element formulation 论混合有限元公式中 inf-sup 条件的必要性
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-02-28 DOI: 10.1093/imanum/drae002
Fleurianne Bertrand, Daniele Boffi
{"title":"On the necessity of the inf-sup condition for a mixed finite element formulation","authors":"Fleurianne Bertrand, Daniele Boffi","doi":"10.1093/imanum/drae002","DOIUrl":"https://doi.org/10.1093/imanum/drae002","url":null,"abstract":"We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $textbf{H}(operatorname{textrm{div}};varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140000943","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
Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier–Stokes equations 不可压缩纳维-斯托克斯方程的连续内部惩罚无发散有限元方法的压力和对流稳健边界
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-02-07 DOI: 10.1093/imanum/drad108
Bosco García-Archilla, Julia Novo
{"title":"Pressure and convection robust bounds for continuous interior penalty divergence-free finite element methods for the incompressible Navier–Stokes equations","authors":"Bosco García-Archilla, Julia Novo","doi":"10.1093/imanum/drad108","DOIUrl":"https://doi.org/10.1093/imanum/drad108","url":null,"abstract":"In this paper, we analyze a pressure-robust method based on divergence-free mixed finite element methods with continuous interior penalty stabilization. The main goal is to prove an $O(h^{k+1/2})$ error estimate for the $L^2$ norm of the velocity in the convection dominated regime. This bound is pressure robust (the error bound of the velocity does not depend on the pressure) and also convection robust (the constants in the error bounds are independent of the Reynolds number).","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705063","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 Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain 演化域中线性化纳维-斯托克斯问题的欧拉有限元法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-30 DOI: 10.1093/imanum/drad105
Michael Neilan, Maxim Olshanskii
{"title":"An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain","authors":"Michael Neilan, Maxim Olshanskii","doi":"10.1093/imanum/drad105","DOIUrl":"https://doi.org/10.1093/imanum/drad105","url":null,"abstract":"The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $L^{2}(H^{1})$-type norm for the pressure component.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644033","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
Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems 基于均衡通量重构的目标导向误差估计，用于涡流问题中谐波近似公式的计算
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-29 DOI: 10.1093/imanum/drad107
Emmanuel Creusé, Serge Nicaise, Zuqi Tang
{"title":"Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems","authors":"Emmanuel Creusé, Serge Nicaise, Zuqi Tang","doi":"10.1093/imanum/drad107","DOIUrl":"https://doi.org/10.1093/imanum/drad107","url":null,"abstract":"In this work, we propose an a posteriori goal-oriented error estimator for the harmonic $textbf {A}$-$varphi $ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Our theoretical results are illustrated by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139577496","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
Cauchy data for Levin’s method 列文方法的柯西数据
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-25 DOI: 10.1093/imanum/drad106
Anthony Ashton
{"title":"Cauchy data for Levin’s method","authors":"Anthony Ashton","doi":"10.1093/imanum/drad106","DOIUrl":"https://doi.org/10.1093/imanum/drad106","url":null,"abstract":"In this paper, we describe the Cauchy data that gives rise to slowly oscillating solutions to the Levin equation. We present a general result on the existence of a unique minimizer of $|Bx|$ subject to the constraint $Ax=y$, where $A,B$ are linear, but not necessarily bounded operators on a complex Hilbert space. This result is used to obtain the solution to the Levin equation, both in the univariate and multivariate case, which minimizes the mean-square of the derivative over the domain. The Cauchy data that generates this solution is then obtained, and this can be used to supplement the Levin equation in the computation of highly oscillatory integrals in the presence of stationary points.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139568229","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 operator preconditioning and the accuracy of solving linear systems 全算子预处理与线性系统求解精度
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-25 DOI: 10.1093/imanum/drad104
Stephan Mohr, Yuji Nakatsukasa, Carolina Urzúa-Torres
{"title":"Full operator preconditioning and the accuracy of solving linear systems","authors":"Stephan Mohr, Yuji Nakatsukasa, Carolina Urzúa-Torres","doi":"10.1093/imanum/drad104","DOIUrl":"https://doi.org/10.1093/imanum/drad104","url":null,"abstract":"Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139568274","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
Well-posedness and error estimates for coupled systems of nonlocal conservation laws 非局部守恒定律耦合系统的好拟性和误差估计
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-20 DOI: 10.1093/imanum/drad101
Aekta Aggarwal, Helge Holden, Ganesh Vaidya
{"title":"Well-posedness and error estimates for coupled systems of nonlocal conservation laws","authors":"Aekta Aggarwal, Helge Holden, Ganesh Vaidya","doi":"10.1093/imanum/drad101","DOIUrl":"https://doi.org/10.1093/imanum/drad101","url":null,"abstract":"This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: (1) Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform $operatorname {BV}$ bound on the numerical approximations; (2) Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; (3) Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505816","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 with rates for a Riccati-based discretization of SLQ problems with SPDEs 基于 Riccati 的 SPDE SLQ 问题离散化的收敛率
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-19 DOI: 10.1093/imanum/drad097
Andreas Prohl, Yanqing Wang
{"title":"Convergence with rates for a Riccati-based discretization of SLQ problems with SPDEs","authors":"Andreas Prohl, Yanqing Wang","doi":"10.1093/imanum/drad097","DOIUrl":"https://doi.org/10.1093/imanum/drad097","url":null,"abstract":"We consider a new discretization in space (parameter $h>0$) and time (parameter $tau>0$) of a stochastic optimal control problem, where a quadratic functional is minimized subject to a linear stochastic heat equation with linear noise. Its construction is based on the perturbation of a generalized difference Riccati equation to approximate the related feedback law. We prove a convergence rate of almost ${mathcal O}(h^{2}+tau )$ for its solution, and conclude from it a rate of almost ${mathcal O}(h^{2}+tau )$ resp. ${mathcal O}(h^{2}+tau ^{1/2})$ for computable approximations of the optimal state and control with additive resp. multiplicative noise.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505922","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
Corrigendum to: Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs 更正：非对称线性椭圆 PDEs 的自适应有限元与准最优总成本
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-01-19 DOI: 10.1093/imanum/drad103
Maximilian Brunner, Michael Innerberger, Ani Miraçi, Dirk Praetorius, Julian Streitberger, Pascal Heid
{"title":"Corrigendum to: Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs","authors":"Maximilian Brunner, Michael Innerberger, Ani Miraçi, Dirk Praetorius, Julian Streitberger, Pascal Heid","doi":"10.1093/imanum/drad103","DOIUrl":"https://doi.org/10.1093/imanum/drad103","url":null,"abstract":"Unfortunately, there is a flaw in the numerical analysis of the published version [IMA J. Numer. Anal., DOI:10.1093/imanum/drad039], which is corrected here. Neither the algorithm nor the results are affected, but constants have to be adjusted.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139505972","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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