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

Authors:M. García-Esnaola; J. M. Peña Abstract: Abstract B-matrices form a subclass of P-matrices for which error bounds for the linear complementarity problem are known. It is proved that a bound involved in such problems is asymptotically optimal. \(B_\pi ^R\) -matrices form a subclass of P-matrices containing B-matrices. For the \(B_\pi ^R\) -matrices, error bounds for the linear complementarity problem are obtained. We also include illustrative examples showing the sharpness of these bounds. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0209-9

Authors:Mehdi Ashkartizabi; Mina Aminghafari; Adel Mohammadpour Abstract: Abstract We find square roots of a complex-valued matrix \(A_{3 \times 3}\) using equation \(B^{2}=A\) . The proposed method is faster than Higham’s method and provides up to 8 square roots with less relative residual and error. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0202-3

Authors:Thomas P. Wihler Abstract: Abstract In this note we shall devise a variable-order continuous Galerkin time stepping method which is especially geared towards norm-preserving dynamical systems. In addition, we will provide an a posteriori estimate for the \(L^\infty \) -error. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0203-2

Authors:Morteza Kimiaei Abstract: Abstract A nonmonotone trust-region method for the solution of nonlinear systems of equations with box constraints is considered. The method differs from existing trust-region methods both in using a new nonmonotonicity strategy in order to accept the current step and a new updating technique for the trust-region-radius. The overall method is shown to be globally convergent. Moreover, when combined with suitable Newton-type search directions, the method preserves the local fast convergence. Numerical results indicate that the new approach is more effective than existing trust-region algorithms. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0208-x

Authors:Beatrice Meini; Tommaso Nesti Abstract: Abstract We consider the problem of solving a rational matrix equation arising in the solution of G-networks. We propose and analyze two numerical methods: a fixed point iteration and the Newton–Raphson method. The fixed point iteration is shown to be globally convergent with linear convergence rate, while the Newton method is shown to have a local convergence, with quadratic convergence rate. Numerical experiments show the effectiveness of the proposed methods. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0214-7

Authors:Stephan Dahlke; Philipp Keding; Thorsten Raasch Abstract: Abstract In the spirit of subatomic or quarkonial decomposition of function spaces (Triebel in Fractals and spectra related to fourier analysis and function spaces. Birkhäuser, Boston, 1997), we construct compactly supported, piecewise polynomial functions whose properly weighted dilates and translates generate frames for Sobolev spaces on the real line. All frame elements except for those on the coarsest level have vanishing moment properties. As a consequence, the matrix representation of certain elliptic operators in frame coordinates is compressible, i.e., well-approximable by sparse submatrices. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0210-3

Authors:Alexandre Ern; Iain Smears; Martin Vohralík Abstract: Abstract We establish the existence of liftings into discrete subspaces of \(\varvec{H}({{\mathrm{div}}})\) of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in the a posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with \(H^{-1}\) source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0217-4

Authors:A. H. Bentbib; M. El Guide; K. Jbilou Abstract: Abstract In the present paper, we propose a new method to inexpensively determine a suitable value of the regularization parameter and an associated approximate solution, when solving ill-conditioned linear system of equations with multiple right-hand sides contaminated by errors. The proposed method is based on the symmetric block Lanczos algorithm, in connection with block Gauss quadrature rules to inexpensively approximate matrix-valued function of the form \(W^Tf(A)W\) , where \(W\in {\mathbb {R}}^{n\times k}\) , \(k\ll n\) , and \(A\in {\mathbb {R}}^{n\times n}\) is a symmetric matrix. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0206-z

Authors:Phung Van Manh Abstract: Abstract We study the asymptotic behavior of harmonic interpolation of harmonic functions based on Radon projections when the chords coalesce to some points, a chord and a point. We show that the limit is the Lagrange or Taylor-type interpolation at coalescing points or chords. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0212-9

Authors:H. J. Wang; D. X. Cao; H. Liu; L. Qiu Abstract: Abstract This paper establishes a numerical validation test for solutions of systems of absolute value equations based on the Poincaré–Miranda theorem. In this paper, the Moore–Kioustelidis theorem is generalized for nondifferential systems of absolute value equations. Numerical results are reported to show the efficiency of the new test method. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0204-1

Authors:Ravi Kanth Adivi Sri Venkata; Murali Mohan Kumar Palli Abstract: Abstract In this paper an exponentially fitted spline method is presented for solving singularly perturbed convection delay problems with boundary layer at left (or right) end of the domain. The error analysis of the scheme is investigated. It is shown that the proposed scheme provides second order accuracy, independent of the perturbation parameter. Numerical results are presented to illustrate the efficiency of the method. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0215-6

Authors:Yongxin Dong; Chuanqing Gu; Zhibing Chen Abstract: Abstract The PageRank algorithm is one of the most commonly used techniques that determines the global importance of Web pages. In this paper, we present a preconditioned Arnoldi-Inout approach for the computation of Pagerank vector, which can take the advantage of both a new two-stage matrix splitting iteration and the Arnoldi process. The implementation and convergence of the new algorithm are discussed in detail. Numerical experiments are presented to illustrate the effectiveness of our approaches. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0211-2

Authors:Mahabub Basha Pathan; Shanthi Vembu Abstract: Abstract In this article, a parameter-uniform hybrid numerical method is presented to solve a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms. Due to these discontinuities, interior layers appear in the solution of the problem considered. The hybrid numerical method uses the standard finite difference scheme in the coarse mesh region and the cubic spline difference scheme in the fine mesh region which is constructed on piecewise-uniform Shishkin mesh. Second order one sided difference approximations are used at the point of discontinuity. Error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to validate the theoretical results. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0218-3

Authors:Yuan-Ming Wang Abstract: Abstract This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0207-y

Authors:Dazhi Zhao; Maokang Luo Abstract: Abstract Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term \(t^{1-\alpha }\) in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0213-8

Authors:Georgios Akrivis; Yiorgos-Sokratis Smyrlis Abstract: Abstract We analyze the discretization of the periodic initial value problem for Kuramoto–Sivashinsky type equations with Burgers nonlinearity by implicit–explicit backward difference formula (BDF) methods, establish stability and derive optimal order error estimates. We also study discretization in space by spectral methods. PubDate: 2017-09-01 DOI: 10.1007/s10092-016-0205-0

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: 2017-09-01 DOI: 10.1007/s10092-017-0220-9

Authors:Ronald DeVore; Guergana Petrova; Przemyslaw Wojtaszczyk Abstract: Abstract This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\) , \(j=1,\ldots ,m\) , where the \(l_j\) are linear functionals from \(\mathcal{X}^*\) . Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\) . Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \) , and for K the unit ball of a smoothness space in \(\mathcal{X}\) . Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\) , with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\) , which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \) . These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation. PubDate: 2017-09-01 DOI: 10.1007/s10092-017-0216-5