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A higher order multiscale method for the wave equation 波方程的高阶多尺度方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-09-19 DOI: 10.1093/imanum/drae059
Felix Krumbiegel, Roland Maier
{"title":"A higher order multiscale method for the wave equation","authors":"Felix Krumbiegel, Roland Maier","doi":"10.1093/imanum/drae059","DOIUrl":"https://doi.org/10.1093/imanum/drae059","url":null,"abstract":"In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher order extension of the localized orthogonal decomposition method combined with a higher order time stepping scheme and present rigorous a priori error estimates in the energy-induced norm. We find that in the very general setting without additional assumptions on the coefficient beyond boundedness arbitrary orders of convergence cannot be expected, but that increasing the polynomial degree may still considerably reduce the size of the error. Under additional regularity assumptions higher orders can be obtained as well. Numerical examples are presented that confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142275655","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 energy stable discrete variational derivative schemes for gradient flows 梯度流的高阶能量稳定离散变分导数方案
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-09-19 DOI: 10.1093/imanum/drae062
Jizu Huang
{"title":"High-order energy stable discrete variational derivative schemes for gradient flows","authors":"Jizu Huang","doi":"10.1093/imanum/drae062","DOIUrl":"https://doi.org/10.1093/imanum/drae062","url":null,"abstract":"The existing discrete variational derivative method is fully implicit and only second-order accurate for gradient flow. In this paper, we propose a framework to construct high-order implicit (original) energy stable schemes and second-order semi-implicit (modified) energy stable schemes. Combined with the Runge–Kutta process, we can build high-order and unconditionally (original) energy stable schemes based on the discrete variational derivative method. The new energy stable schemes are implicit and leads to a large sparse nonlinear algebraic system at each time step, which can be efficiently solved by using an inexact Newton-type solver. To avoid solving nonlinear algebraic systems, we then present a relaxed discrete variational derivative method, which can construct second-order, linear and unconditionally (modified) energy stable schemes. Several numerical simulations are performed to investigate the efficiency, stability and accuracy of the newly proposed schemes.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142275656","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 mesh-independent method for second-order potential mean field games 二阶势均场博弈的网格无关方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-09-18 DOI: 10.1093/imanum/drae061
Kang Liu, Laurent Pfeiffer
{"title":"A mesh-independent method for second-order potential mean field games","authors":"Kang Liu, Laurent Pfeiffer","doi":"10.1093/imanum/drae061","DOIUrl":"https://doi.org/10.1093/imanum/drae061","url":null,"abstract":"This article investigates the convergence of the Generalized Frank–Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142245188","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
Low regularity error estimates for the time integration of 2D NLS 二维 NLS 时间积分的低正则误差估计
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-09-13 DOI: 10.1093/imanum/drae054
Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz
{"title":"Low regularity error estimates for the time integration of 2D NLS","authors":"Lun Ji, Alexander Ostermann, Frédéric Rousset, Katharina Schratz","doi":"10.1093/imanum/drae054","DOIUrl":"https://doi.org/10.1093/imanum/drae054","url":null,"abstract":"A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $mathbb{T}^{2}$. The scheme is analysed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^{s}(mathbb{T}^{2})$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $tau ^{s/2}$ in $L^{2}(mathbb{T}^{2})$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142231539","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 mini immersed finite element method for two-phase Stokes problems on Cartesian meshes 笛卡尔网格上两相斯托克斯问题的微型沉浸式有限元方法
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-09-09 DOI: 10.1093/imanum/drae053
Haifeng Ji, Dong Liang, Qian Zhang
{"title":"A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes","authors":"Haifeng Ji, Dong Liang, Qian Zhang","doi":"10.1093/imanum/drae053","DOIUrl":"https://doi.org/10.1093/imanum/drae053","url":null,"abstract":"This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160426","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
New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations 基于巴拿赫空间的新混合有限元方法，用于耦合孔弹性方程和热方程
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-09-05 DOI: 10.1093/imanum/drae052
Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier
{"title":"New Banach spaces-based mixed finite element methods for the coupled poroelasticity and heat equations","authors":"Julio Careaga, Gabriel N Gatica, Cristian Inzunza, Ricardo Ruiz-Baier","doi":"10.1093/imanum/drae052","DOIUrl":"https://doi.org/10.1093/imanum/drae052","url":null,"abstract":"In this paper, we introduce and analyze a Banach spaces-based approach yielding a fully-mixed finite element method for numerically solving the coupled poroelasticity and heat equations, which describe the interaction between the fields of deformation and temperature. A nonsymmetric pseudostress tensor is utilized to redefine the constitutive equation for the total stress, which is an extension of Hooke’s law to account for thermal effects. The resulting continuous formulation, posed in suitable Banach spaces, consists of a coupled system of three saddle point-type problems, each with right-hand terms that depend on data and the unknowns of the other two. The well-posedness of it is analyzed by means of a fixed-point strategy, so that the classical Banach theorem, along with the Babuška–Brezzi theory in Banach spaces, allows to conclude, under a smallness assumption on the data, the existence of a unique solution. The discrete analysis is conducted in a similar manner, utilizing the Brouwer and Banach theorems to demonstrate both the existence and uniqueness of the discrete solution. The rates of convergence of the resulting Galerkin method are then presented. Finally, a number of numerical tests are shown to validate the aforementioned statement and demonstrate the good performance of the method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142142448","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
Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes 在简单网格上避免巴布什卡悖论的必要条件和充分条件
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-08-27 DOI: 10.1093/imanum/drae050
Sören Bartels, Philipp Tscherner
{"title":"Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes","authors":"Sören Bartels, Philipp Tscherner","doi":"10.1093/imanum/drae050","DOIUrl":"https://doi.org/10.1093/imanum/drae050","url":null,"abstract":"It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084952","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
Computing Klein-Gordon Spectra 计算克莱因-戈登频谱
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-08-26 DOI: 10.1093/imanum/drae032
Frank Rösler, Christiane Tretter
{"title":"Computing Klein-Gordon Spectra","authors":"Frank Rösler, Christiane Tretter","doi":"10.1093/imanum/drae032","DOIUrl":"https://doi.org/10.1093/imanum/drae032","url":null,"abstract":"We study the computational complexity of the eigenvalue problem for the Klein–Gordon equation in the framework of the Solvability Complexity Index Hierarchy. We prove that the eigenvalue of the Klein–Gordon equation with linearly decaying potential can be computed in a single limit with guaranteed error bounds from above. The proof is constructive, i.e. we obtain a numerical algorithm that can be implemented on a computer. Moreover, we prove abstract enclosures for the point spectrum of the Klein–Gordon equation and we compare our numerical results to these enclosures. Finally, we apply both the implemented algorithm and our abstract enclosures to several physically relevant potentials such as Sauter and cusp potentials and we provide a convergence and error analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084953","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
Mixed finite elements for the Gross–Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound 格罗斯-皮塔耶夫斯基特征值问题的混合有限元：先验误差分析和有保证的能量下限
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-08-22 DOI: 10.1093/imanum/drae048
Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim
{"title":"Mixed finite elements for the Gross–Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound","authors":"Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim","doi":"10.1093/imanum/drae048","DOIUrl":"https://doi.org/10.1093/imanum/drae048","url":null,"abstract":"We establish an a priori error analysis for the lowest-order Raviart–Thomas finite element discretization of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conforming approaches, which naturally imply upper energy bounds, the proposed mixed discretization provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142042431","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
Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations 不可压缩纳维-斯托克斯方程的质量、动量和能量保全 FEEC 和破碎 FEEC 方案
IF 2.1 2区 数学
IMA Journal of Numerical Analysis Pub Date : 2024-08-15 DOI: 10.1093/imanum/drae047
Valentin Carlier, Martin Campos Pinto, Francesco Fambri
{"title":"Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations","authors":"Valentin Carlier, Martin Campos Pinto, Francesco Fambri","doi":"10.1093/imanum/drae047","DOIUrl":"https://doi.org/10.1093/imanum/drae047","url":null,"abstract":"In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141991902","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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