Authors:Silvia Bertoluzza; Micol Pennacchio; Daniele Prada 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: 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

Authors:M. García-Esnaola; J. M. Peña 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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: 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