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A unified immersed finite element error analysis for one-dimensional interface problems 一维界面问题的统一沉浸式有限元误差分析
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-29 DOI: 10.1007/s10543-024-01014-z
Slimane Adjerid, Tao Lin, Haroun Meghaichi
{"title":"A unified immersed finite element error analysis for one-dimensional interface problems","authors":"Slimane Adjerid, Tao Lin, Haroun Meghaichi","doi":"10.1007/s10543-024-01014-z","DOIUrl":"https://doi.org/10.1007/s10543-024-01014-z","url":null,"abstract":"<p>It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001645","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
Admissible subspaces and the subspace iteration method 可容许子空间和子空间迭代法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-28 DOI: 10.1007/s10543-024-01012-1
{"title":"Admissible subspaces and the subspace iteration method","authors":"","doi":"10.1007/s10543-024-01012-1","DOIUrl":"https://doi.org/10.1007/s10543-024-01012-1","url":null,"abstract":"<h3>Abstract</h3> <p>In this work we revisit the convergence analysis of the Subspace Iteration Method (SIM) for the computation of approximations of a matrix <em>A</em> by matrices of rank <em>h</em>. Typically, the analysis of convergence of these low-rank approximations has been obtained by first estimating the (angular) distance between the subspaces produced by the SIM and the dominant subspaces of <em>A</em>. It has been noticed that this approach leads to upper bounds that overestimate the approximation error in case the <em>h</em>th singular value of <em>A</em> lies in a cluster of singular values. To overcome this difficulty we introduce a substitute for dominant subspaces, which we call admissible subspaces. We develop a proximity analysis of subspaces produced by the SIM to admissible subspaces; in turn, this analysis allows us to obtain novel estimates for the approximation error by low-rank matrices obtained by the implementation of the deterministic SIM. Our results apply in the case when the <em>h</em>th singular value of <em>A</em> belongs to a cluster of singular values. Indeed, our approach allows us to consider the case when the <em>h</em>th and the <span> <span>((h+1))</span> </span>st singular values of <em>A</em> coincide, which does not seem to be covered by previous works in the deterministic setting.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140001840","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
Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions 二维封闭曲线上近奇异线积分的奇异交换正交
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-27 DOI: 10.1007/s10543-024-01013-0
Ludvig af Klinteberg
{"title":"Singularity swap quadrature for nearly singular line integrals on closed curves in two dimensions","authors":"Ludvig af Klinteberg","doi":"10.1007/s10543-024-01013-0","DOIUrl":"https://doi.org/10.1007/s10543-024-01013-0","url":null,"abstract":"<p>This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently was introduced for boundaries discretized using composite Gauss–Legendre quadrature. The original method builds on swapping the target singularity for its preimage in the complexified space of the curve parametrization, where the source panel is flat. This allows the integral to be efficiently evaluated using an interpolatory quadrature with a monomial basis. In this extension, we use the target preimage to swap the singularity to a point close to the unit circle. This allows us to evaluate the integral using an interpolatory quadrature with complex exponential basis functions. This is well-conditioned, and can be efficiently evaluated using the fast Fourier transform. The resulting method has exponential convergence, and can be used to accurately evaluate layer potentials close to the source geometry. We report experimental results on a simple test geometry, and provide a baseline Julia implementation that can be used for further experimentation.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139979315","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
Structured eigenvalue backward errors for rational matrix functions with symmetry structures 具有对称结构的有理矩阵函数的结构化特征值后向误差
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-17 DOI: 10.1007/s10543-024-01010-3
Anshul Prajapati, Punit Sharma
{"title":"Structured eigenvalue backward errors for rational matrix functions with symmetry structures","authors":"Anshul Prajapati, Punit Sharma","doi":"10.1007/s10543-024-01010-3","DOIUrl":"https://doi.org/10.1007/s10543-024-01010-3","url":null,"abstract":"<p>We derive computable formulas for the structured backward errors of a complex number <span>(lambda )</span> when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, <span>(*)</span>-palindromic, T-even, T-odd, <span>(*)</span>-even, and <span>(*)</span>-odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139764879","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
Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations 弱奇异 Volterra 积分微分方程的分数配位法的收敛性和超收敛性
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-12 DOI: 10.1007/s10543-024-01011-2
{"title":"Convergence and superconvergence of a fractional collocation method for weakly singular Volterra integro-differential equations","authors":"","doi":"10.1007/s10543-024-01011-2","DOIUrl":"https://doi.org/10.1007/s10543-024-01011-2","url":null,"abstract":"<h3>Abstract</h3> <p>A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time <span> <span>(t=0)</span> </span>. A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765222","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 convolution quadrature using derivatives and its application 使用导数的卷积正交及其应用
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-09 DOI: 10.1007/s10543-024-01009-w
Hao Ren, Junjie Ma, Huilan Liu
{"title":"A convolution quadrature using derivatives and its application","authors":"Hao Ren, Junjie Ma, Huilan Liu","doi":"10.1007/s10543-024-01009-w","DOIUrl":"https://doi.org/10.1007/s10543-024-01009-w","url":null,"abstract":"<p>This paper is devoted to explore the convolution quadrature based on a class of two-point Hermite collocation methods. Incorporating derivatives into the numerical scheme enhances the accuracy while preserving stability, which is confirmed by the convergence analysis for the discretization of the initial value problem. Moreover, we employ the resulting quadrature to evaluate a class of highly oscillatory integrals. The frequency-explicit convergence analysis demonstrates that the proposed convolution quadrature surpasses existing convolution quadratures, achieving the highest convergence rate with respect to the oscillation among them. Numerical experiments involving convolution integrals with smooth, weakly singular, and highly oscillatory Bessel kernels illustrate the reliability and efficiency of the proposed convolution quadrature.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139765022","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 posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems 奇异扰动反应扩散问题二元有限元法的后验误差估计
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-02-05 DOI: 10.1007/s10543-024-01008-x
JaEun Ku, Martin Stynes
{"title":"A posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems","authors":"JaEun Ku, Martin Stynes","doi":"10.1007/s10543-024-01008-x","DOIUrl":"https://doi.org/10.1007/s10543-024-01008-x","url":null,"abstract":"<p><i>A posteriori</i> error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139689193","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
Improved uniform error bounds on parareal exponential algorithm for highly oscillatory systems 高振荡系统准指数算法的改进均匀误差边界
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-01-31 DOI: 10.1007/s10543-023-01005-6
Bin Wang, Yaolin Jiang
{"title":"Improved uniform error bounds on parareal exponential algorithm for highly oscillatory systems","authors":"Bin Wang, Yaolin Jiang","doi":"10.1007/s10543-023-01005-6","DOIUrl":"https://doi.org/10.1007/s10543-023-01005-6","url":null,"abstract":"<p>For the well known parareal algorithm, we formulate and analyse a novel class of parareal exponential schemes with improved uniform accuracy for highly oscillatory system <span>(ddot{q}+frac{1}{varepsilon ^2}M q =frac{1}{varepsilon ^{mu }}f(q))</span> with <span>(mu =0)</span> or 1. The solution of this considered system propagates waves with wavelength at <span>(mathcal {O} (varepsilon ))</span> in time and the value of <span>(mu )</span> corresponds to the strength of nonlinearity. This brings significantly numerical burden in scientific computation for highly oscillatory systems with <span>(0&lt;varepsilon ll 1)</span>. The new proposed algorithm is formulated by using some reformulation approaches to the problem, Fourier pseudo-spectral methods, and parareal exponential integrators. The fast Fourier transform is incorporated in the implementation. We rigorously study the convergence, showing that for nonlinear systems, the algorithm has improved uniform accuracy <span>(mathcal {O}big ( varepsilon ^{(2k+3)(1-mu )}Delta t^{2k+2}+varepsilon ^{5(1-mu )}delta t^4big ))</span> in the position and <span>(mathcal {O}big ( varepsilon ^{(2k+3)(1-mu )-1}Delta t^{2k+2}+varepsilon ^{4-5mu }delta t^4big ))</span> in the momenta, where <i>k</i> is the number of parareal iterations, and <span>(Delta t)</span> and <span>(delta t)</span> are two time stepsizes used in the algorithm. The energy conservation is also discussed and the algorithm is shown to have an improved energy conservation. Numerical experiments are provided and the numerical results demonstrate the improved uniform accuracy and improved energy conservation of the obtained integrator through four Hamiltonian differential equations including nonlinear wave equations.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139648530","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
On the stability radius for linear time-delay systems 论线性时延系统的稳定半径
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-01-30 DOI: 10.1007/s10543-023-01006-5
{"title":"On the stability radius for linear time-delay systems","authors":"","doi":"10.1007/s10543-023-01006-5","DOIUrl":"https://doi.org/10.1007/s10543-023-01006-5","url":null,"abstract":"<h3>Abstract</h3> <p>The exponential function that appears in the formula of the stability radius of linear time-delay differential systems is approximated by its Padé approximant. This reduces the computation of the level sets of singular values in the stability radius formula to the computation of imaginary eigenvalues of special matrix polynomials. Then a bisection method is used for computing lower and upper bounds on the stability radius. A rounding error analysis is presented. Several numerical examples are given to demonstrate the feasibility and efficiency of the bisection method.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139648091","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
Incremental algorithms for truncated higher-order singular value decompositions 截断高阶奇异值分解的增量算法
IF 1.5 3区 数学
BIT Numerical Mathematics Pub Date : 2024-01-08 DOI: 10.1007/s10543-023-01004-7
Chao Zeng, Michael K. Ng, Tai-Xiang Jiang
{"title":"Incremental algorithms for truncated higher-order singular value decompositions","authors":"Chao Zeng, Michael K. Ng, Tai-Xiang Jiang","doi":"10.1007/s10543-023-01004-7","DOIUrl":"https://doi.org/10.1007/s10543-023-01004-7","url":null,"abstract":"<p>We develop and study incremental algorithms for truncated higher-order singular value decompositions. By combining the SVD updating and different truncated higher-order singular value decompositions, two incremental algorithms are proposed. Not only the factor matrices but also the core tensor are updated in an incremental style. The costs of these algorithms are compared and the approximation errors are analyzed. Numerical results demonstrate that the proposed incremental algorithms have advantages in online computation.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139398101","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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