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  • Articles  (257)
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  • Articles  (257)
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  • 11
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    Optimal vorticity accuracy in an efficient velocity–vorticity method for the 2D Navier–Stokes equations (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time \(H^1\) stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity , thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 12
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    Some error analysis on virtual element methods (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: Some error analyses on virtual element methods (VEMs) including inverse inequalities, norm equivalence, and interpolation error estimates are developed for polygonal meshes, each element of which admits a virtual quasi-uniform triangulation. This sub-mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are presented by means of standard technical tools in finite element methods.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 13
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    Improved convergence theorems of modulus-based matrix splitting iteration method for nonlinear complementarity problems of H -matrices (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: In this paper, the convergence conditions of the modulus-based matrix splitting iteration method for nonlinear complementarity problem of H -matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones. Numerical examples show the advantages of the new theorems.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 14
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    An inexact Newton method for solving complementarity problems in hydrodynamic lubrication (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: We present an iterative procedure based on a damped inexact Newton iteration for solving linear complementarity problems. We introduce the method in the framework of a popular problem arising in mechanical engineering: the analysis of cavitation in lubricated contacts. In this context, we show how the perturbation and the damping parameter are chosen in our method and we prove the global convergence of the entire procedure. A Fortran implementation of the method is finally analyzed. First, we validate the procedure and analyze all its components, performing also a comparison with a recently proposed technique based on the Fischer–Burmeister–Newton iteration. Then, we solve a 2D problem and provide some insights on an efficient implementation of the method exploiting routines of the Lapack and of the PETSc packages for the solution of inner linear systems.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 15
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    Critical lengths of cycloidal spaces are zeros of Bessel functions (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: Cycloidal spaces are generated by the trigonometric polynomials of degree one and algebraic polynomials. The critical length of a cycloidal space is the supremum of the lengths of the intervals on which the Hermite interpolation problems are unisolvent. The critical length is related with the critical length for design purposes in computer-aided geometric design. This paper shows an unexpected connection of critical lengths with the zeros of Bessel functions. We prove that the half of the critical length of a cycloidal space is the first positive zero of a Bessel function of the first kind.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 16
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    Image reconstruction from scattered Radon data by weighted positive definite kernel functions (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 17
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    Comparison results for K -nonnegative double splittings of K -monotone matrices (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: This paper introduces the K -nonnegative double splitting of a K -monotone matrix using knowledge of the matrices that leave a cone \(K\subseteq \mathbb {R}^n\) invariant. The convergence of this splitting is studied. Comparison theorems for two K -nonnegative double splittings of a K -monotone matrix are obtained. The results generalize the corresponding results introduced by Song and Song (Calcolo 48:245–260, 2011 ) for nonnegative double splitting. Some examples are provided to illustrate the main results.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 18
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    A real structure-preserving method for the quaternion LU decomposition, revisited (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: In a paper published in 2013, Wang and Ma proposed a structure-preserving algorithm for computing the quaternion LU decomposition. They claimed that it was faster than the LU decomposition implemented in the quaternion Toolbox for Matlab (QTFM). But in 2015, Sangwine, one of the authors of QTFM, pointed out that the tests carried out by him did not support Wang and Ma’s claim. We studied the structure-preserving algorithm of Wang and Ma, and found that the computations were based on element to element operations. In this paper, we re-propose real structure-preserving methods for the quaternion LU decomposition and partial pivoting quaternion LU decomposition, which make full use of high-level operations, and relation of operations between quaternion matrices and their real representation matrices. These algorithms are more efficient than that in QTFM using quaternion arithmetics. Numerical experiments are provided to demonstrate the efficiency of the real structure-preserving method.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 19
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    A phase-field model for liquid–gas mixtures: mathematical modelling and discontinuous Galerkin discretization (2018)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2018-03-06
    Description: To model a liquid–gas mixture, in this article we propose a phase-field approach that might also provide a description of the expansion stage of a metal foam inside a hollow mold. We conceive the mixture as a two-phase incompressible–compressible fluid governed by a Navier–Stokes–Cahn–Hilliard system of equations, and we adapt the Lowengrub–Truskinowsky model to take into account the expansion of the gaseous phase. The resulting system of equations is characterized by a velocity field that fails to be divergence-free, by a logarithmic term for the pressure that enters in the Gibbs free-energy expression and by the viscosity that degenerates in the gas phase. In the second part of the article we propose an energy-based numerical scheme that, at the discrete level, preserves the mass conservation property and the energy dissipation law of the original system. We use a discontinuous Galerkin approximation for the spatial approximation and a modified midpoint based scheme for the time approximation.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 20
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    On the nonmonotonicity degree of nonmonotone line searches (2017)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2017-05-13
    Description: The nonmonotone globalization technique is useful in difficult nonlinear problems, because of the fact that it may help escaping from steep sided valleys and may improve both the possibility of finding the global optimum and the rate of convergence. This paper discusses the nonmonotonicity degree of nonmonotone line searches for the unconstrained optimization. Specifically, we analyze some popular nonmonotone line search methods and explore, from a computational point of view, the relations between the efficiency of a nonmonotone line search and its nonmonotonicity degree. We attempt to answer this question how to control the degree of the nonmonotonicity of line search rules in order to reach a more efficient algorithm. Hence in an attempt to control the nonmonotonicity degree, two adaptive nonmonotone rules based on the morphology of the objective function are proposed. The global convergence and the convergence rate of the proposed methods are analysed under mild assumptions. Numerical experiments are made on a set of unconstrained optimization test problems of the CUTEr (Gould et al. in ACM Trans Math Softw 29:373–394, 2003 ) collection. The performance data are first analysed through the performance profile of Dolan and Moré (Math Program 91:201–213, 2002 ). In the second kind of analyse, the performance data are analysed in terms of increasing dimension of the test problems.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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