Authors:Donatella Occorsio; Giada Serafini Abstract: The paper deals with the approximation of integrals of the type $$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$ where \({\mathrm {D}}=[-\,1,1]^2\) , f is a function defined on \({\mathrm {D}}\) with possible algebraic singularities on \(\partial {\mathrm {D}}\) , \({\mathbf {w}}\) is the product of two Jacobi weight functions, and the kernel \({\mathbf {K}}\) can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed. PubDate: 2018-02-07 DOI: 10.1007/s10092-018-0243-x Issue No:Vol. 55, No. 1 (2018)

Authors:M. Akbas; L. G. Rebholz; C. Zerfas Abstract: 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. PubDate: 2018-02-07 DOI: 10.1007/s10092-018-0246-7 Issue No:Vol. 55, No. 1 (2018)

Authors:Long Chen; Jianguo Huang Abstract: 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. PubDate: 2018-02-07 DOI: 10.1007/s10092-018-0249-4 Issue No:Vol. 55, No. 1 (2018)

Authors:S. De Marchi; A. Iske; G. Santin Abstract: 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. PubDate: 2018-02-02 DOI: 10.1007/s10092-018-0247-6 Issue No:Vol. 55, No. 1 (2018)

Authors:Francesco Mezzadri; Emanuele Galligani Abstract: 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. PubDate: 2018-02-01 DOI: 10.1007/s10092-018-0244-9 Issue No:Vol. 55, No. 1 (2018)

Authors:Hanyu Li; Shaoxin Wang Pages: 1121 - 1146 Abstract: In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the data space and the solution space are measured by the weighted Frobenius norm and the Euclidean norm, respectively. Then, we investigate the corresponding structured partial condition number when the problem is structured. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical examples. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0221-8 Issue No:Vol. 54, No. 4 (2017)

Authors:Jie Ma; Linlin Qi; Yongshu Li Pages: 1147 - 1168 Abstract: In this paper, we derive novel representations of generalized inverses \(A^{(1)}_{T,S}\) and \(A^{(1,2)}_{T,S}\) , which are much simpler than those introduced in Ben-Israel and Greville (Generalized inverses: theory and applications. Springer, New York, 2003). When \(A^{(1,2)}_{T,S}\) is applied to matrices of index one, a simple representation for the group inverse \(A_{g}\) is derived. Based on these representations, we derive various algorithms for computing \(A^{(1)}_{T,S}\) , \(A^{(1,2)}_{T,S}\) and \(A_{g}\) , respectively. Moreover, our methods can be achieved through Gauss–Jordan elimination and complexity analysis indicates that our method for computing the group inverse \(A_{g}\) is more efficient than the other existing methods in the literature for a large class of problems in the computational complexity sense. Finally, numerical experiments show that our method for the group inverse \(A_{g}\) has highest accuracy among all the existing methods in the literature and also has the lowest cost of CPU time when applied to symmetric matrices or matrices with high rank or small size matrices with low rank in practice. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0222-7 Issue No:Vol. 54, No. 4 (2017)

Authors:Veselina K. Kyncheva; Viktor V. Yotov; Stoil I. Ivanov Pages: 1199 - 1212 Abstract: In this paper, we establish a general theorem for iteration functions in a cone normed space over \({{\mathbb {R}}}^n\) . Using this theorem together with a general convergence theorem of Proinov (J Complex 33:118–144, 2016), we obtain a local convergence theorem with a priori and a posteriori error estimates as well as a theorem under computationally verifiable initial conditions for the Schröder’s iterative method considered as a method for simultaneous computation of polynomial zeros of unknown multiplicity. Numerical examples which demonstrate the convergence properties of the proposed method are also provided. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0225-4 Issue No:Vol. 54, No. 4 (2017)

Authors:J. K. Liu; S. J. Li Pages: 1213 - 1215 Abstract: In this note, we show that the proof of Remark 3 of Lemma 3.2 in “A three-term derivative-free projection method for nonlinear monotone system of equations” (Calcolo 53:427–450, 2016) is not correct, which implies that the conclusion of Remark 3 is not appropriate to prove Theorem 3.1. A new proof of Remark 3 is established, which guarantees the corresponding global convergence Theorem 3.1. Throughout, we use the same notations and equation numbers as in the above reference. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0224-5 Issue No:Vol. 54, No. 4 (2017)

Authors:Á. P. Horváth Pages: 1265 - 1291 Abstract: Chromatic derivatives and series expansions of bandlimited functions have recently been introduced in signal processing and they have been shown to be useful in practical applications. We extend the notion of chromatic derivative using varying weights. When the kernel function of the integral operator is positive, this extension ensures chromatic expansions around every point. Besides old examples, the modified method is demonstrated via some new ones as Walsh–Fourier transform, and Poisson-wavelet transform. Moreover the convergence of the de la Vallée Poussin means of chromatic series expansions in weighted \(L^p\) -spaces is investigated. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0229-0 Issue No:Vol. 54, No. 4 (2017)

Authors:Caixun Wang Pages: 1293 - 1303 Abstract: 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. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0230-7 Issue No:Vol. 54, No. 4 (2017)

Authors:P. Boito; Y. Eidelman; L. Gemignani Pages: 1305 - 1338 Abstract: We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an \(N\times N\) structured matrix pencil using O(N) flops per iteration and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0231-6 Issue No:Vol. 54, No. 4 (2017)

Authors:Elisabetta Repossi; Riccardo Rosso; Marco Verani Pages: 1339 - 1377 Abstract: 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. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0233-4 Issue No:Vol. 54, No. 4 (2017)

Authors:Carlos García; Gabriel N. Gatica; Antonio Márquez; Salim Meddahi Pages: 1419 - 1439 Abstract: We propose an implicit Newmark method for the time integration of the pressure–stress formulation of a fluid–structure interaction problem. The space Galerkin discretization is based on the Arnold–Falk–Winther mixed finite element method with weak symmetry in the solid and the usual Lagrange finite element method in the acoustic medium. We prove that the resulting fully discrete scheme is well-posed and uniformly stable with respect to the discretization parameters and Poisson ratio, and we provide asymptotic error estimates. Finally, we present numerical tests to confirm the asymptotic error estimates predicted by the theory. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0234-3 Issue No:Vol. 54, No. 4 (2017)

Authors:Buyang Li Pages: 1441 - 1480 Abstract: In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg–Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform \(L^{3+\delta }\) regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a \(\ell ^2(W^{1,3+\delta })\) estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0237-0 Issue No:Vol. 54, No. 4 (2017)

Authors:Ying Li; Musheng Wei; Fengxia Zhang; Jianli Zhao Abstract: 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. PubDate: 2017-11-03 DOI: 10.1007/s10092-017-0241-4

Authors:Emmanuil H. Georgoulis; Tristan Pryer Abstract: We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control. PubDate: 2017-11-01 DOI: 10.1007/s10092-017-0240-5

Authors:J. M. Carnicer; E. Mainar; J. M. Peña Abstract: 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. PubDate: 2017-10-04 DOI: 10.1007/s10092-017-0239-y

Authors:Hua Zheng Abstract: 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. PubDate: 2017-09-05 DOI: 10.1007/s10092-017-0236-1

Authors:M. Álvarez; G. N. Gatica; R. Ruiz-Baier Abstract: We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fully-mixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68–95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart–Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems. PubDate: 2017-09-05 DOI: 10.1007/s10092-017-0238-z