Authors:Wei Ma Pages: 503 - 514 Abstract: Abstract In this paper, we present the explicit and computable formula of the structured backward errors of the generalized saddle point systems. Simple numerical examples show that the expressions are useful for testing the stability of practical algorithms. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0195-y Issue No:Vol. 54, No. 2 (2017)

Authors:F. Z. Geng; S. P. Qian Pages: 515 - 526 Abstract: Abstract In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing kernel method. Two numerical examples are given to show the effectiveness of the present method. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0196-x Issue No:Vol. 54, No. 2 (2017)

Authors:Sukhjit Singh; D. K. Gupta; Rakesh P. Badoni; E. Martínez; José L. Hueso Pages: 527 - 539 Abstract: Abstract The local convergence analysis of a parameter based iteration with Hölder continuous first derivative is studied for finding solutions of nonlinear equations in Banach spaces. It generalizes the local convergence analysis under Lipschitz continuous first derivative. The main contribution is to show the applicability to those problems for which Lipschitz condition fails without using higher order derivatives. An existence-uniqueness theorem along with the derivation of error bounds for the solution is established. Different numerical examples including nonlinear Hammerstein equation are solved. The radii of balls of convergence for them are obtained. Substantial improvements of these radii are found in comparison to some other existing methods under similar conditions for all examples considered. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0197-9 Issue No:Vol. 54, No. 2 (2017)

Authors:Dmitriy Leykekhman; Buyang Li Pages: 541 - 565 Abstract: Abstract As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon–Miranda) discrete maximum principle, and then prove the stability of the Ritz projection with mixed boundary conditions in \(L^\infty \) norm. Such results have a number of applications, but are not available in the literature. Our proof of the maximum-norm stability of the Ritz projection is based on converting the mixed boundary value problem to a pure Neumann problem, which is of independent interest. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0198-8 Issue No:Vol. 54, No. 2 (2017)

Authors:Giuseppe Mastroianni; Gradimir V. Milovanović; Incoronata Notarangelo Pages: 567 - 585 Abstract: Abstract A class of Fredholm integral equations of the second kind, with respect to the exponential weight function \(w(x)=\exp (-(x^{-\alpha }+x^\beta ))\) , \(\alpha >0\) , \(\beta >1\) , on \((0,+\infty )\) , is considered. The kernel k(x, y) and the function g(x) in such kind of equations, $$\begin{aligned} f(x)-\mu \int _0^{+\infty }k(x,y)f(y)w(y)\mathrm {d}y =g(x),\quad x\in (0,+\infty ), \end{aligned}$$ can grow exponentially with respect to their arguments, when they approach to \(0^+\) and/or \(+\infty \) . We propose a simple and suitable Nyström-type method for solving these equations. The study of the stability and the convergence of this numerical method in based on our results on weighted polynomial approximation and “truncated” Gaussian rules, recently published in Mastroianni and Notarangelo (Acta Math Hung, 142:167–198, 2014), and Mastroianni, Milovanović and Notarangelo (IMA J Numer Anal 34:1654–1685, 2014) respectively. Moreover, we prove a priori error estimates and give some numerical examples. A comparison with other Nyström methods is also included. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0199-7 Issue No:Vol. 54, No. 2 (2017)

Authors:Jeonghun J. Lee Pages: 587 - 607 Abstract: Abstract We propose mixed finite element methods for the standard linear solid model in viscoelasticity and prove a priori error estimates. In our mixed formulation the governing equations of the problem become a symmetric hyperbolic system, so we can use standard techniques for a priori error estimates and time discretization. Numerical results illustrating our theoretical analysis are included. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0200-5 Issue No:Vol. 54, No. 2 (2017)

Authors:M. Akhmouch; M. Benzakour Amine Pages: 609 - 641 Abstract: Abstract In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The numerical approximation makes use of a standard finite volume scheme in space with a special treatment for the convection-diffusion fluxes which are approximated by the classical Il’in fluxes. For the time discretization, we introduce our linearized semi-exponentially fitted scheme. The paper gives a comparison between the proposed scheme and different versions of linearized backward Euler schemes. The existence and uniqueness of a numerical solution to the scheme and its convergence to a weak solution of the studied system are proved. In the last section, we present some numerical tests to show the performance of our method. Our numerical approach is then applied to a chemotaxis-growth model describing bacterial pattern formation. PubDate: 2017-06-01 DOI: 10.1007/s10092-016-0201-4 Issue No:Vol. 54, No. 2 (2017)

Authors:M. H. Annaby; M. M. Tharwat Pages: 23 - 41 Abstract: Abstract We implement the sinc method to compute the eigenvalues of a second order boundary value problem with mixed type boundary conditions where the eigenparameter appears linearly in the boundary conditions. We investigate the behavior of the solutions as well as the characteristic determinant via successive iterations. The method is implemented by splitting the characteristic determinant into two parts, where it is proved that the unknown one lies in a Paley-Wiener space and it is approximated by an interpolation sampling theorem. Examples are illustrated numerically and graphically. PubDate: 2017-03-01 DOI: 10.1007/s10092-016-0174-3 Issue No:Vol. 54, No. 1 (2017)

Authors:Bin Wang; Hongli Yang; Fanwei Meng Pages: 117 - 140 Abstract: Abstract This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge–Kutta–Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. PubDate: 2017-03-01 DOI: 10.1007/s10092-016-0179-y Issue No:Vol. 54, No. 1 (2017)

Authors:D. Irisarri Pages: 141 - 154 Abstract: Abstract In this paper, we present a methodology for stabilizing the virtual element method applied to the convection-diffusion-reaction equation. The stabilization is carried out modifying the mesh inside the boundary layer so that the link-cutting condition is satisfied. The method provides a stable solution to all regimes. Numerical examples are presented for several regimes in which satisfactory results are obtained. PubDate: 2017-03-01 DOI: 10.1007/s10092-016-0180-5 Issue No:Vol. 54, No. 1 (2017)

Authors:Hossein Aminikhah; Javad Alavi Pages: 299 - 317 Abstract: Abstract The B-spline collocation methods and a new ODEs solver based on B-spline quasi-interpolation are developed to study the problem of forced convection over a horizontal flat plate, numerically. The problem is a system of nonlinear ordinary differential equations which arises in boundary layer flow. A more accurate value of \(\sigma =f^{\prime \prime }(0)\) obtained by applying quartic B-spline collocation method and utilized to solve the system of ODE. The results are shown to be precise as compared to the corresponding results obtained by Howarth. PubDate: 2017-03-01 DOI: 10.1007/s10092-016-0188-x Issue No:Vol. 54, No. 1 (2017)

Authors:Michael Feischl; Thomas Führer; Dirk Praetorius; Ernst P. Stephan Pages: 367 - 399 Abstract: Abstract For the non-preconditioned Galerkin matrix of the hypersingular integral operator, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on adaptively refined meshes, require the development of appropriate preconditioners. We propose and analyze a local multilevel preconditioner which is optimal in the sense that the condition number of the corresponding preconditioned system is independent of the number of elements, the local mesh-size, and the number of refinement levels. The theory covers closed boundaries as well as open screens in 2D and 3D. Numerical experiments underline the analytical results and compare the proposed preconditioner to other multilevel schemes as well as techniques based on operator preconditioning. PubDate: 2017-03-01 DOI: 10.1007/s10092-016-0190-3 Issue No:Vol. 54, No. 1 (2017)

Authors:Paraskevi Fika; Marilena Mitrouli Pages: 455 - 470 Abstract: Abstract In the present work we study how the Aitken’s method can be applied to the sequence of moments \(\{c_k=(x, A^{k}x),~k \in {\mathbb {Z}}\},\) of a given symmetric positive definite matrix \(A \in {\mathbb {R}}^{p \times p},\) for the prediction of possible unknown terms of this sequence. Direct estimation of \(c_k\) leads to approximating bilinear quantities i.e. \((y, A^{k}x)\) as well. By employing Taylor series expansion for appropriate f, prediction of \(y^Tf(A)x\) can be achieved. Estimates for many useful linear algebra quantities can be derived by appropriately selecting the vectors y and x. Numerical examples concerning such applications are presented. The estimates are illustrated through numerical examples executed on the high-performance computing system ARIS. PubDate: 2017-03-01 DOI: 10.1007/s10092-016-0193-0 Issue No:Vol. 54, No. 1 (2017)

Authors:Hadi Nosratipour; Akbar Hashemi Borzabadi; Omid Solaymani Fard Abstract: Abstract 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. PubDate: 2017-05-12 DOI: 10.1007/s10092-017-0226-3

Authors:J. K. Liu; S. J. Li 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: 2017-05-09 DOI: 10.1007/s10092-017-0224-5

Authors:Veselina K. Kyncheva; Viktor V. Yotov; Stoil I. Ivanov 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: 2017-05-04 DOI: 10.1007/s10092-017-0225-4

Authors:P. F. Antonietti; P. Houston; X. Hu; M. Sarti; M. Verani 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: 2017-05-04 DOI: 10.1007/s10092-017-0223-6

Authors:Jie Ma; Linlin Qi; Yongshu Li 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: 2017-04-09 DOI: 10.1007/s10092-017-0222-7

Authors:Hanyu Li; Shaoxin Wang 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: 2017-04-07 DOI: 10.1007/s10092-017-0221-8

Authors:Micol Ferranti; Bruno Iannazzo; Thomas Mach; Raf Vandebril Abstract: Abstract An extended QR algorithm specifically tailored for Hamiltonian matrices is presented. The algorithm generalizes the customary Hamiltonian QR algorithm with additional freedom in choosing between various possible extended Hamiltonian Hessenberg forms. We introduced in Ferranti et al. (Calcolo, 2015. doi:10.1007/s10092-016-0192-1) an algorithm to transform certain Hamiltonian matrices to such forms. Whereas the convergence of the classical QR algorithm is related to classical Krylov subspaces, convergence in the extended case links to extended Krylov subspaces, resulting in a greater flexibility, and possible enhanced convergence behavior. Details on the implementation, covering the bidirectional chasing and the bulge exchange based on rotations are presented. The numerical experiments reveal that the convergence depends on the selected extended forms and illustrate the validity of the approach. PubDate: 2017-03-08 DOI: 10.1007/s10092-017-0220-9