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On the injectivity of mean value mappings between quadrilaterals 关于四边形间均值映射的注入性
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-07-10 DOI: 10.1007/s10444-025-10246-8
Michael S. Floater, Georg Muntingh
{"title":"On the injectivity of mean value mappings between quadrilaterals","authors":"Michael S. Floater,&nbsp;Georg Muntingh","doi":"10.1007/s10444-025-10246-8","DOIUrl":"10.1007/s10444-025-10246-8","url":null,"abstract":"<div><p>Mean value coordinates can be used to map one polygon into another, with application to computer graphics and curve and surface modelling. In this paper, we show that if the polygons are quadrilaterals, and if the target quadrilateral is convex, then the mapping is injective.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10246-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A SUPG-stabilized virtual element method for the Navier–Stokes equation: approximations of branches of non-singular solutions Navier-Stokes方程的supg稳定虚元法：非奇异解分支的逼近
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-07-07 DOI: 10.1007/s10444-025-10247-7
Sudheer Mishra, E. Natarajan, Sundararajan Natarajan
{"title":"A SUPG-stabilized virtual element method for the Navier–Stokes equation: approximations of branches of non-singular solutions","authors":"Sudheer Mishra,&nbsp;E. Natarajan,&nbsp;Sundararajan Natarajan","doi":"10.1007/s10444-025-10247-7","DOIUrl":"10.1007/s10444-025-10247-7","url":null,"abstract":"<div><p>In this paper, we investigate a stabilization technique for the Navier–Stokes equations for incompressible fluid flow using equal-order virtual element pairs on general polygonal meshes. We propose a residual-based SUPG-like stabilization term to address the violation of the discrete inf-sup condition, which leads to pressure instability, and to mitigate the effects of the convection-dominated regime. Additionally, we employ a grad-div stabilization term to address the violation of divergence-free constraints. We extend the concept of nonlinear stability derived in (López-Marcos and Sanz-Serna, IMA J. Numer. Anal. <b>8</b>(1), 71–84, 1998) to a stabilized virtual element framework. Following the results of Lopez-Marcos &amp; Sanz-Serna, we establish the well-posedness and optimal convergence estimates in the energy norm using the branches of non-singular solutions. We perform several numerical experiments to validate the theoretical findings.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Nonconforming virtual element method for general second-order elliptic problems on curved domain 曲面上一般二阶椭圆问题的非协调虚元法
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-07-04 DOI: 10.1007/s10444-025-10242-y
Yi Liu, Alessandro Russo
{"title":"Nonconforming virtual element method for general second-order elliptic problems on curved domain","authors":"Yi Liu,&nbsp;Alessandro Russo","doi":"10.1007/s10444-025-10242-y","DOIUrl":"10.1007/s10444-025-10242-y","url":null,"abstract":"<div><p>The nonconforming virtual element method with curved edges was proposed and analyzed for the Poisson equation by L. Beirão da Veiga, Y. Liu, L. Mascotto, and A. Russo in (J. Sci. Comput. <b>99</b>(1) 2024). The goal of this paper is to extend the nonconforming virtual element method to a more general second-order elliptic problem with variable coefficients in domains with curved boundaries and curved internal interfaces. We prove an optimal convergence of arbitrary order in the energy and <span>(L^2)</span>-norms, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the method is shown to be comparable with that obtained from the theoretical analysis.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 orthonormal gradient flow for computing ground state solution of two-dimensional dipolar fermion gas 计算二维偶极费米子气体基态解的正交梯度流
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-07-04 DOI: 10.1007/s10444-025-10248-6
Xuelin Zhang, Hanquan Wang
{"title":"An orthonormal gradient flow for computing ground state solution of two-dimensional dipolar fermion gas","authors":"Xuelin Zhang,&nbsp;Hanquan Wang","doi":"10.1007/s10444-025-10248-6","DOIUrl":"10.1007/s10444-025-10248-6","url":null,"abstract":"<div><p>In this paper, based on density functional theory, we present an orthonormal gradient flow (OGF) for finding the ground state solution of a two-dimensional dipolar fermion gas. The OGF has the properties of orthonormality preserving and energy diminishing. By evolving such OGF, we may get the ground state solution of the dipolar fermion gas numerically. The OGF consists of time-dependent integral and partial differential equations. In principle, it can be discretized with many kinds of numerical techniques. We propose a backward Euler Fourier spectral method to discretize such OGF numerically. Numerical tests are reported to demonstrate the effectiveness of the proposed methods. The proposed numerical methods are applied to compute the ground state solution of the ultracold dipolar fermion gas.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Solving elliptic optimal control problems via neural networks and optimality system 利用神经网络和最优系统求解椭圆型最优控制问题
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-06-26 DOI: 10.1007/s10444-025-10241-z
Yongcheng Dai, Bangti Jin, Ramesh Chandra Sau, Zhi Zhou
{"title":"Solving elliptic optimal control problems via neural networks and optimality system","authors":"Yongcheng Dai,&nbsp;Bangti Jin,&nbsp;Ramesh Chandra Sau,&nbsp;Zhi Zhou","doi":"10.1007/s10444-025-10241-z","DOIUrl":"10.1007/s10444-025-10241-z","url":null,"abstract":"<div><p>In this work, we investigate a neural network-based solver for optimal control problems (without/with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order optimality system of the optimal control problem and employs deep neural networks to represent the solutions to the reduced system. We present an error analysis of the scheme and provide <span>(L^2(Omega ))</span> error bounds on the state, control, and adjoint in terms of neural network parameters (e.g., depth, width, and parameter bounds) and the numbers of sampling points. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the method and compare it with two existing ones.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 nonconforming P3+B4 and discontinuous P2 mixed finite element on tetrahedral grids 四面体网格上不一致P3+B4和不连续P2混合有限元
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-06-23 DOI: 10.1007/s10444-025-10244-w
Xuejun Xu, Shangyou Zhang
{"title":"A nonconforming P3+B4 and discontinuous P2 mixed finite element on tetrahedral grids","authors":"Xuejun Xu,&nbsp;Shangyou Zhang","doi":"10.1007/s10444-025-10244-w","DOIUrl":"10.1007/s10444-025-10244-w","url":null,"abstract":"<div><p>A nonconforming <span>(P_3)</span> finite element is constructed by enriching the conforming <span>(P_3)</span> finite element space with nine <span>(P_4)</span> nonconforming bubbles, on each tetrahedron. Here, the divergence of the <span>(P_4)</span> bubble is not a <span>(P_3)</span> polynomial, but a <span>(P_2)</span> polynomial. This nonconforming <span>(P_3)</span> finite element, combined with the discontinuous <span>(P_2)</span> finite element, is inf-sup stable for solving the Stokes equations on general tetrahedral grids. Consequently, such a mixed finite element method produces quasi-optimal solutions for solving the stationary Stokes equations. With these special <span>(P_4)</span> bubbles, the discrete velocity remains locally pointwise divergence-free. Numerical tests confirm the theory.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Decoupled weak Galerkin finite element method for Maxwell’s equations 麦克斯韦方程组的解耦弱伽辽金有限元法
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-06-20 DOI: 10.1007/s10444-025-10243-x
Wenya Qi, Kaifang Liu
{"title":"Decoupled weak Galerkin finite element method for Maxwell’s equations","authors":"Wenya Qi,&nbsp;Kaifang Liu","doi":"10.1007/s10444-025-10243-x","DOIUrl":"10.1007/s10444-025-10243-x","url":null,"abstract":"<div><p>We consider Maxwell’s equations in a decoupled formulation by introducing Lagrange multipliers and obtain the magnetic field given the known electric field. The proposed formulation combines the decoupled weak form with the four equations of Maxwell’s model. The decoupled system reduces the computational complexity by restricting the degrees of freedom of the electric or magnetic fields. We present the construction of mixed weak Galerkin finite element methods for electric field and magnetic field, utilizing backward Euler time discretization in fully discrete schemes. We analyze the error estimate of the electric and magnetic field in the energy norm. Finally, we present numerical results for the proposed schemes in three-dimensional space to validate our theory.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 4","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
From completeness of discrete translates to phaseless sampling of the short-time Fourier transform 从离散的完全性转换到短时傅里叶变换的无相采样
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-06-13 DOI: 10.1007/s10444-025-10236-w
Philipp Grohs, Lukas Liehr, Irina Shafkulovska
{"title":"From completeness of discrete translates to phaseless sampling of the short-time Fourier transform","authors":"Philipp Grohs,&nbsp;Lukas Liehr,&nbsp;Irina Shafkulovska","doi":"10.1007/s10444-025-10236-w","DOIUrl":"10.1007/s10444-025-10236-w","url":null,"abstract":"<div><p>We study the uniqueness problem in short-time Fourier transform phase retrieval by exploring a connection to the completeness problem of discrete translates. Specifically, we prove that functions in <span>( L^2(K) )</span> with <span>( K subseteq {{mathbb {R}}^d})</span> compact, are uniquely determined by phaseless lattice-samples of its short-time Fourier transform with window function <i>g</i>, provided that specific density properties of translates of <i>g</i> are met. By proving completeness statements for systems of discrete translates in Banach function spaces on compact sets, we obtain new uniqueness statements for phaseless sampling on lattices beyond the known Gaussian window regime. Our results apply to a large class of window functions which are relevant in time-frequency analysis and applications.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 3","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-025-10236-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Obstructions for Gabor frames of the second-order B-spline 二阶b样条Gabor框架的障碍物
IF 2.1 3区 数学
Advances in Computational Mathematics Pub Date : 2025-06-05 DOI: 10.1007/s10444-025-10239-7
Riya Ghosh, A. Antony Selvan
{"title":"Obstructions for Gabor frames of the second-order B-spline","authors":"Riya Ghosh,&nbsp;A. Antony Selvan","doi":"10.1007/s10444-025-10239-7","DOIUrl":"10.1007/s10444-025-10239-7","url":null,"abstract":"<div><p>For a window <span>( gin L^2(mathbb {R}) )</span>, the subset of all lattice parameters <span>( (a, b)in mathbb {R}^2_+ )</span> such that <span>( mathcal {G}(g,a,b)={e^{2pi ib mcdot }g(cdot -a k): k, min mathbb {Z}} )</span> forms a frame for <span>( L^2(mathbb {R}) )</span> is known as the frame set of <i>g</i>. In time-frequency analysis, determining the Gabor frame set for a given window is a challenging open problem. In particular, the frame set for B-splines has many obstructions. Lemvig and Nielsen in (J. Fourier Anal. Appl. <b>22</b>, 1440–1451, 2016) conjectured that if </p><div><div><span>$$begin{aligned} a_0=dfrac{1}{2m+1},~ b_0=dfrac{2k+1}{2},~k,min mathbb {N},~k&gt;m,~a_0b_0&lt;1, end{aligned}$$</span></div></div><p>then the Gabor system <span>( mathcal {G}(Q_2, a, b) )</span> of the second-order B-spline <span>( Q_2 )</span> is not a frame along the hyperbolas </p><div><div><span>$$begin{aligned} ab=dfrac{2k+1}{2(2m+1)},text { for }bin left[ b_0-a_0dfrac{k-m}{2}, b_0+a_0dfrac{k-m}{2}right] , end{aligned}$$</span></div></div><p>for every <span>( a_0 )</span>, <span>( b_0 )</span>. Nielsen in (2015) also conjectured that <span>( mathcal {G}(Q_2, a,b) )</span> is not a frame for </p><div><div><span>$$a=dfrac{1}{2m},~b=dfrac{2k+1}{2},~k,min mathbb {N},~k&gt;m,~ab&lt;1text { with }gcd (4m,2k+1)=1.$$</span></div></div><p>In this paper, we prove that both Conjectures are true.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 3","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144924559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Noniterative localized exponential time differencing methods for hyperbolic conservation laws 双曲型守恒律的非迭代局域指数差分方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2025-05-27 DOI: 10.1007/s10444-025-10240-0
Cao-Kha Doan, Phuoc-Toan Huynh, Thi-Thao-Phuong Hoang
{"title":"Noniterative localized exponential time differencing methods for hyperbolic conservation laws","authors":"Cao-Kha Doan,&nbsp;Phuoc-Toan Huynh,&nbsp;Thi-Thao-Phuong Hoang","doi":"10.1007/s10444-025-10240-0","DOIUrl":"10.1007/s10444-025-10240-0","url":null,"abstract":"<div><p>The paper is concerned with efficient time discretization methods based on exponential integrators for scalar hyperbolic conservation laws. The model problem is first discretized in space by the discontinuous Galerkin method, resulting in a system of nonlinear ordinary differential equations. To solve such a system, exponential time differencing of order 2 (ETDRK2) is employed with Jacobian linearization at each time step. The scheme is fully explicit and relies on the computation of matrix exponential vector products. To accelerate such computation, we further construct a noniterative, nonoverlapping domain decomposition algorithm, namely localized ETDRK2, which loosely decouples the system at each time step via suitable interface conditions. Temporal error analysis of the proposed global and localized ETDRK2 schemes is rigorously proved; moreover, the schemes are shown to be conservative under periodic boundary conditions. Numerical results for the Burgers’ equation in one and two dimensions (with moving shocks) are presented to verify the theoretical results and illustrate the performance of the global and localized ETDRK2 methods where large time step sizes can be used without affecting numerical stability.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"51 3","pages":""},"PeriodicalIF":1.7,"publicationDate":"2025-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144140275","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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