Numerische Mathematik最新文献

The pressure-wired Stokes element: a mesh-robust version of the Scott–Vogelius element 压力有线斯托克斯元素：斯科特-沃格柳斯元素的网格稳健版

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-08-24 DOI: 10.1007/s00211-024-01430-x

Benedikt Gräßle, Nis-Erik Bohne, Stefan Sauter

{"title":"The pressure-wired Stokes element: a mesh-robust version of the Scott–Vogelius element","authors":"Benedikt Gräßle, Nis-Erik Bohne, Stefan Sauter","doi":"10.1007/s00211-024-01430-x","DOIUrl":"https://doi.org/10.1007/s00211-024-01430-x","url":null,"abstract":"The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order (k-1). It employs a “singular distance” (measured by some geometric mesh quantity ( Theta left( textbf{z}right) ge 0) for triangle vertices (textbf{z})) and imposes a local side condition on the pressure space associated to vertices (textbf{z}) with (Theta left( textbf{z}right) =0). The method is inf-sup stable for any fixed regular triangulation and (kge 4). However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices (0<Theta left( textbf{z}right) ll 1). In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the “singular distance”. We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142222656","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

Runge–Kutta convolution quadrature based on Gauss methods 基于高斯方法的 Runge-Kutta 卷积正交法

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-08-10 DOI: 10.1007/s00211-024-01429-4

L. Banjai, M. Ferrari

引用次数: 0

Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations 非均质不可压缩纳维-斯托克斯方程有限差分法的数学分析

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-07-25 DOI: 10.1007/s00211-024-01421-y

Kohei Soga

引用次数: 0

Which constraints of a numerical problem cause ill-conditioning? 数值问题的哪些约束条件会导致条件不符？

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-07-16 DOI: 10.1007/s00211-024-01427-6

Nick Dewaele, Nick Vannieuwenhoven

引用次数: 0

A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws 参数相关双曲守恒定律熵解的矩方法

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-07-15 DOI: 10.1007/s00211-024-01428-5

Clément Cardoen, Swann Marx, Anthony Nouy, Nicolas Seguin

{"title":"A moment approach for entropy solutions of parameter-dependent hyperbolic conservation laws","authors":"Clément Cardoen, Swann Marx, Anthony Nouy, Nicolas Seguin","doi":"10.1007/s00211-024-01428-5","DOIUrl":"https://doi.org/10.1007/s00211-024-01428-5","url":null,"abstract":"We propose a numerical method to solve parameter-dependent scalar hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre’s hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge. In the end, several post-treatments can be performed from this approximate moments sequence. In particular, the graph of the solution can be reconstructed from an optimization of the Christoffel–Darboux kernel associated with the approximate measure, that is a powerful approximation tool able to capture a large class of irregular functions. Also, for uncertainty quantification problems, several quantities of interest can be estimated, sometimes directly such as the expectation of smooth functionals of the solutions. The performance of our approach is evaluated through numerical experiments on the inviscid Burgers equation with parametrised initial conditions or parametrised flux function.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719225","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

Circuits of ferromagnetic nanowires 铁磁纳米线电路

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-07-04 DOI: 10.1007/s00211-024-01426-7

Sergiy M. Bokoch, Gilles Carbou, Stéphane Labbé, Stéphane Despréaux

引用次数: 0

Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states 用高斯相干态高效逼近高频亥姆霍兹解法

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-07-04 DOI: 10.1007/s00211-024-01411-0

T. Chaumont-Frelet, V. Dolean, M. Ingremeau

引用次数: 0

Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine-Gordon equation 用于正弦-戈登方程长时动力学的劳森型指数积分器的改进均匀误差边界

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-07-02 DOI: 10.1007/s00211-024-01423-w

Yue Feng, Katharina Schratz

{"title":"Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine-Gordon equation","authors":"Yue Feng, Katharina Schratz","doi":"10.1007/s00211-024-01423-w","DOIUrl":"https://doi.org/10.1007/s00211-024-01423-w","url":null,"abstract":"We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is (O(varepsilon )) with (0 < varepsilon ll 1) a dimensionless parameter up to the time at (O(1/varepsilon ^2)). The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discretization at (O(varepsilon ^2tau )) instead of (O(tau )) according to classical error estimates and at (O(h^m+varepsilon ^2tau )) for the full-discretization up to the time (T_{varepsilon } = T/varepsilon ^2) with (T>0) fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the nonlinear Klein–Gordon equation with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with (O(varepsilon ^2)) wavelength in time and (O(varepsilon ^{-2})) wave speed, which indicates that the temporal error is independent of (varepsilon ) when the time step size is chosen as (O(varepsilon ^2)). Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526992","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

Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains Lipschitz 域上拉普拉斯-狄利克特问题的强制第二类边界积分方程

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-06-18 DOI: 10.1007/s00211-024-01424-9

S. N. Chandler-Wilde, E. A. Spence

{"title":"Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains","authors":"S. N. Chandler-Wilde, E. A. Spence","doi":"10.1007/s00211-024-01424-9","DOIUrl":"https://doi.org/10.1007/s00211-024-01424-9","url":null,"abstract":"We present new second-kind integral-equation formulations of the interior and exterior Dirichlet problems for Laplace’s equation. The operators in these formulations are both continuous and coercive on general Lipschitz domains in (mathbb {R}^d), (dge 2), in the space (L^2(Gamma )), where (Gamma ) denotes the boundary of the domain. These properties of continuity and coercivity immediately imply that (1) the Galerkin method converges when applied to these formulations; and (2) the Galerkin matrices are well-conditioned as the discretisation is refined, without the need for operator preconditioning (and we prove a corresponding result about the convergence of GMRES). The main significance of these results is that it was recently proved (see Chandler-Wilde and Spence in Numer Math 150(2):299–371, 2022) that there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which the operators in the standard second-kind integral-equation formulations for Laplace’s equation (involving the double-layer potential and its adjoint) cannot be written as the sum of a coercive operator and a compact operator in the space (L^2(Gamma )). Therefore there exist 2- and 3-d Lipschitz domains and 3-d star-shaped Lipschitz polyhedra for which Galerkin methods in ({L^2(Gamma )}) do not converge when applied to the standard second-kind formulations, but do converge for the new formulations.\u0000","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503107","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

$$H(textrm{div})$$ -conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equations 用于斯托克斯方程和纳维-斯托克斯方程的应力-速度公式的 $$H(textrm{div})$$ -conforming HDG 方法

IF 2.1 2区数学

Numerische Mathematik Pub Date : 2024-06-17 DOI: 10.1007/s00211-024-01419-6

Weifeng Qiu, Lina Zhao

{"title":"$$H(textrm{div})$$ -conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equations","authors":"Weifeng Qiu, Lina Zhao","doi":"10.1007/s00211-024-01419-6","DOIUrl":"https://doi.org/10.1007/s00211-024-01419-6","url":null,"abstract":"In this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and (H(textrm{div};Omega ))-conforming space of order (k+1), respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete (H^1)-stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the (L^2) projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete (H^1)-norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete (H^1)-norm is achieved. Several numerical experiments are carried out to confirm the proposed theories.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503105","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