Authors:Hanyu Li; Shaoxin Wang Pages: 1121 - 1146 Abstract: 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: 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:P. F. Antonietti; P. Houston; X. Hu; M. Sarti; M. Verani Pages: 1169 - 1198 Abstract: Abstract In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0223-6 Issue No:Vol. 54, No. 4 (2017)

Authors:Veselina K. Kyncheva; Viktor V. Yotov; Stoil I. Ivanov Pages: 1199 - 1212 Abstract: 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: 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:Constantin Christof Pages: 1243 - 1264 Abstract: Abstract In this paper, we present an alternative approach to a priori \(L^\infty \) -error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0228-1 Issue No:Vol. 54, No. 4 (2017)

Authors:Á. P. Horváth Pages: 1265 - 1291 Abstract: 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: 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: 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: 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:Wei Shi; Kai Liu; Xinyuan Wu; Changying Liu Pages: 1379 - 1402 Abstract: Abstract In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0232-5 Issue No:Vol. 54, No. 4 (2017)

Authors:Shangyou Zhang; Shuo Zhang Pages: 1403 - 1417 Abstract: Abstract This paper presents a procedure to construct stable \(C_0P_2{-}P_0\) finite element pair for three dimensional incompressible Stokes problem. It is proved that, the quadratic-constant finite element pair, though not stable in general, is uniformly stable on a certain family of tetrahedral grids, namely some kind of sub-hexahedron tetrahedral grids. The sub-hexahedron tetrahedral grid is defined by refining each eight-vertex hexahedron of a certain hexahedral grid into twelve tetrahedra with one added vertex inside the hexahedron, while the hexahedral grid is a partition of a polyhedral domain where each (non-flat face) hexahedron is defined by a tri-linear mapping on the unit cube with eight vertices. PubDate: 2017-12-01 DOI: 10.1007/s10092-017-0235-2 Issue No:Vol. 54, No. 4 (2017)

Authors:Carlos García; Gabriel N. Gatica; Antonio Márquez; Salim Meddahi Pages: 1419 - 1439 Abstract: 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: 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:Silvia Bertoluzza; Micol Pennacchio; Daniele Prada Abstract: Abstract We build and analyze balancing domain decomposition by constraint and finite element tearing and interconnecting dual primal preconditioners for elliptic problems discretized by the virtual element method. We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments confirm the theory. PubDate: 2017-11-10 DOI: 10.1007/s10092-017-0242-3

Authors:Ying Li; Musheng Wei; Fengxia Zhang; Jianli Zhao Abstract: 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: 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: 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: 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: 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