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  • Articles  (257)
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  • Articles  (257)
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  • 1
    Publication Date: 2018-03-06
    Description: In this paper, we propose an extended block Krylov process to construct two biorthogonal bases for the extended Krylov subspaces \(\mathbb {K}_{m}^e(A,V)\) and \(\mathbb {K}_{m}^e(A^{T},W)\) , where \(A \in \mathbb {R}^{n \times n}\) and \(V,~W \in \mathbb {R}^{n \times p}\) . After deriving some new theoretical results and algebraic properties, we apply the proposed algorithm with moment matching techniques for model reduction in large scale dynamical systems. Numerical experiments for large and sparse problems are given to show the efficiency of the proposed method.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 2
    Publication Date: 2018-03-06
    Description: In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k -order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\) -order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k -order convergent for the stress in \(H(\mathrm {div})\) -norm and for the displacement in \(L^2\) -norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 3
    Publication Date: 2018-03-06
    Description: 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.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 4
    Publication Date: 2018-03-06
    Description: 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.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 5
    Publication Date: 2018-03-06
    Description: 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.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 6
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    Springer
    In: CALCOLO
    Publication Date: 2018-03-06
    Description: 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.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 7
    Publication Date: 2018-03-06
    Description: 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.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 8
    Publication Date: 2018-03-06
    Description: In this paper, we introduce implicit and explicit iterative methods for finding a zero of a monotone variational inclusion in Hilbert spaces. As consequence, an improvement modification of an algorithm existing in literature is obtained. A numerical example is given for illustrating our algorithm.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 9
    Publication Date: 2018-03-06
    Description: We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler–Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the Matlab ODE suite. Our results indicate that the performance of the Euler–Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 10
    Publication Date: 2018-03-06
    Description: In this paper, we provide three types of general convergence theorems for Picard iteration in n -dimensional vector spaces over a valued field. These theorems can be used as tools to study the convergence of some particular Picard-type iterative methods. As an application, we present a new semilocal convergence theorem for the one-dimensional Newton method for approximating all the zeros of a polynomial simultaneously. This result improves in several directions the previous one given by Batra (BIT Numer Math 42:467–476, 2002 ).
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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