arXiv - MATH - Numerical Analysis最新文献

Application of a Fourier-Type Series Approach based on Triangles of Constant Width to Letterforms 基于恒定宽度三角形的傅立叶级数方法在字母字体中的应用

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11958

Micha Wasem, Florence Yerly

引用次数: 0

Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals 具有凸密度函数的周期性科恩-沙姆方程能量的完全保证和可计算误差边界

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11769

Andrea BordignonCERMICS, Geneviève DussonLMB, Éric CancèsCERMICS, MATHERIALS, Gaspard KemlinLAMFA, Rafael Antonio Lainez ReyesIANS, Benjamin StammIANS

{"title":"Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals","authors":"Andrea BordignonCERMICS, Geneviève DussonLMB, Éric CancèsCERMICS, MATHERIALS, Gaspard KemlinLAMFA, Rafael Antonio Lainez ReyesIANS, Benjamin StammIANS","doi":"arxiv-2409.11769","DOIUrl":"https://doi.org/arxiv-2409.11769","url":null,"abstract":"In this article, we derive fully guaranteed error bounds for the energy of\u0000convex nonlinear mean-field models. These results apply in particular to\u0000Kohn-Sham equations with convex density functionals, which includes the reduced\u0000Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact\u0000exchange-density functional (which is unfortunately not explicit and therefore\u0000not usable in practice). We then decompose the obtained bounds into two parts,\u0000one depending on the chosen discretization and one depending on the number of\u0000iterations performed in the self-consistent algorithm used to solve the\u0000nonlinear eigenvalue problem, paving the way for adaptive refinement\u0000strategies. The accuracy of the bounds is demonstrated on a series of test\u0000cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated\u0000with the rHF model and discretized with planewaves. We also show that, although\u0000not anymore guaranteed, the error bounds remain very accurate for a Silicon\u0000crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation\u0000functionals of practical interest.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Spline-based solution transfer for space-time methods in 2D+t 基于样条的 2D+t 时空方法求解转移

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11639

Logan Larose, Jude T. Anderson, David M. Williams

引用次数: 0

A novel Mortar Method Integration using Radial Basis Functions 使用径向基函数的新型砂浆法积分法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11735

Daniele Moretto, Andrea Franceschini, Massimiliano Ferronato

{"title":"A novel Mortar Method Integration using Radial Basis Functions","authors":"Daniele Moretto, Andrea Franceschini, Massimiliano Ferronato","doi":"arxiv-2409.11735","DOIUrl":"https://doi.org/arxiv-2409.11735","url":null,"abstract":"Recent advancements in computational capabilities have significantly enhanced\u0000the numerical simulation of complex multiphysics and multidomain problems.\u0000However, mesh generation remains a primary bottleneck in these simulations. To\u0000address this challenge, non-conforming grids are often utilized, which\u0000necessitates the development of robust and efficient intergrid interpolator\u0000operators. This paper presents a novel approach for transferring variable\u0000fields across non-conforming meshes within a mortar framework, where weak\u0000continuity conditions are imposed. The key contribution of our work is the\u0000introduction of an innovative algorithm that utilizes Radial Basis Function\u0000(RBF) interpolations to compute the mortar integral, offering a compelling\u0000alternative to traditional projection-based algorithms. Pairing RBF methods\u0000with numerical integration techniques, we propose an efficient algorithm\u0000tailored for complex three-dimensional scenarios. This paper details the\u0000formulation, analysis, and validation of the proposed RBF algorithm through a\u0000series of numerical examples, demonstrating its effectiveness. Furthermore, the\u0000details of the implementation are discussed and a test case involving a complex\u0000geometry is presented, to illustrate the applicability and advantages of our\u0000approach in addressing real-world problems.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials 用于评估层势和体积势的轻量级、几何灵活的快速算法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11998

Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter

{"title":"A Lightweight, Geometrically Flexible Fast Algorithm for the Evaluation of Layer and Volume Potentials","authors":"Fredrik Fryklund, Leslie Greengard, Shidong Jiang, Samuel Potter","doi":"arxiv-2409.11998","DOIUrl":"https://doi.org/arxiv-2409.11998","url":null,"abstract":"Over the last two decades, several fast, robust, and high-order accurate\u0000methods have been developed for solving the Poisson equation in complicated\u0000geometry using potential theory. In this approach, rather than discretizing the\u0000partial differential equation itself, one first evaluates a volume integral to\u0000account for the source distribution within the domain, followed by solving a\u0000boundary integral equation to impose the specified boundary conditions. Here,\u0000we present a new fast algorithm which is easy to implement and compatible with\u0000virtually any discretization technique, including unstructured domain\u0000triangulations, such as those used in standard finite element or finite volume\u0000methods. Our approach combines earlier work on potential theory for the heat\u0000equation, asymptotic analysis, the nonuniform fast Fourier transform (NUFFT),\u0000and the dual-space multilevel kernel-splitting (DMK) framework. It is\u0000insensitive to flaws in the triangulation, permitting not just nonconforming\u0000elements, but arbitrary aspect ratio triangles, gaps and various other\u0000degeneracies. On a single CPU core, the scheme computes the solution at a rate\u0000comparable to that of the fast Fourier transform (FFT) in work per gridpoint.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Adaptive Time-Step Semi-Implicit One-Step Taylor Scheme for Stiff Ordinary Differential Equations 刚性常微分方程的自适应时步法半隐式一步泰勒方案

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11990

S. Boscarino, E. Macca

引用次数: 0

Conditions aux limites fortement non lin{é}aires pour les {é}quations d'Euler de la dynamique des gaz 气体动力学欧拉{e}方程的强非线性边界条件

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.11774

François DuboisLMO, LMSSC

引用次数: 0

A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity 超导静态金兹堡-朗道方程的多尺度方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-18 DOI: arxiv-2409.12023

Christian Döding, Benjamin Dörich, Patrick Henning

{"title":"A multiscale approach to the stationary Ginzburg-Landau equations of superconductivity","authors":"Christian Döding, Benjamin Dörich, Patrick Henning","doi":"arxiv-2409.12023","DOIUrl":"https://doi.org/arxiv-2409.12023","url":null,"abstract":"In this work, we study the numerical approximation of minimizers of the\u0000Ginzburg-Landau free energy, a common model to describe the behavior of\u0000superconductors under magnetic fields. The unknowns are the order parameter,\u0000which characterizes the density of superconducting charge carriers, and the\u0000magnetic vector potential, which allows to deduce the magnetic field that\u0000penetrates the superconductor. Physically important and numerically challenging\u0000are especially settings which involve lattices of quantized vortices which can\u0000be formed in materials with a large Ginzburg-Landau parameter $kappa$. In\u0000particular, $kappa$ introduces a severe mesh resolution condition for\u0000numerical approximations. In order to reduce these computational restrictions,\u0000we investigate a particular discretization which is based on mixed meshes where\u0000we apply a Lagrange finite element approach for the vector potential and a\u0000localized orthogonal decomposition (LOD) approach for the order parameter. We\u0000justify the proposed method by a rigorous a-priori error analysis (in $L^2$ and\u0000$H^1$) in which we keep track of the influence of $kappa$ in all error\u0000contributions. This allows us to conclude $kappa$-dependent resolution\u0000conditions for the various meshes and which only impose moderate practical\u0000constraints compared to a conventional finite element discretization. Finally,\u0000our theoretical findings are illustrated by numerical experiments.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions GePUP-ES：具有无滑动条件的不可压缩纳维-斯托克斯方程的高阶能量稳定投影方法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.11255

Yang Li, Xu Wu, Jiatu Yan, Jiang Yang, Qinghai Zhang, Shubo Zhao

{"title":"GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions","authors":"Yang Li, Xu Wu, Jiatu Yan, Jiang Yang, Qinghai Zhang, Shubo Zhao","doi":"arxiv-2409.11255","DOIUrl":"https://doi.org/arxiv-2409.11255","url":null,"abstract":"Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007\u0000Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP\u0000formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving\u0000the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this\u0000paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a)\u0000electric boundary conditions with no explicit enforcement of the no-penetration\u0000condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the\u0000divergence of an initially non-solenoidal velocity, and (d) monotonic decrease\u0000of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES\u0000formulations are of strong forms and are designed for finite volume/difference\u0000methods under the framework of method of lines. Furthermore, we develop\u0000semi-discrete algorithms that preserve (c) and (d) and fully discrete\u0000algorithms that are fourth-order accurate for velocity both in time and in\u0000space. These algorithms employ algebraically stable time integrators in a\u0000black-box manner and only consist of solving a sequence of linear equations in\u0000each time step. Results of numerical tests confirm our analysis.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Adaptively Coupled Domain Decomposition Method for Multiphase and Multicomponent Porous Media Flows 多相和多组分多孔介质流动的自适应耦合领域分解法

arXiv - MATH - Numerical Analysis Pub Date : 2024-09-17 DOI: arxiv-2409.10875

Shizhe Li, Li Zhao, Chen-Song Zhang

{"title":"Adaptively Coupled Domain Decomposition Method for Multiphase and Multicomponent Porous Media Flows","authors":"Shizhe Li, Li Zhao, Chen-Song Zhang","doi":"arxiv-2409.10875","DOIUrl":"https://doi.org/arxiv-2409.10875","url":null,"abstract":"Numerical simulation of large-scale multiphase and multicomponent flow in\u0000porous media is a significant field of interest in the petroleum industry. The\u0000fully implicit approach is favored in reservoir simulation due to its numerical\u0000stability and relaxed constraints on time-step sizes. However, this method\u0000requires solving a large nonlinear system at each time step, making the\u0000development of efficient and convergent numerical methods crucial for\u0000accelerating the nonlinear solvers. In this paper, we present an adaptively\u0000coupled subdomain framework based on the domain decomposition method. The\u0000solution methods developed within this framework effectively handle strong\u0000nonlinearities in global problems by addressing subproblems in the coupled\u0000regions. Furthermore, we propose several adaptive coupling strategies and\u0000develop a method for leveraging initial guesses to accelerate the solution of\u0000nonlinear problems, thereby improving the convergence and parallel performance\u0000of nonlinear solvers. A series of numerical experiments validate the\u0000effectiveness of the proposed framework. Additionally, by utilizing tens of\u0000thousands of processors, we demonstrate the scalability of this approach\u0000through a large-scale reservoir simulation with over 2 billion degrees of\u0000freedom.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0