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  • Articles  (146)
  • 2010-2014  (146)
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  • Articles  (146)
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  • 11
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    A posteriori error estimates for nonconforming streamline-diffusion finite element methods for convection-diffusion problems (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-07-23
    Description: We consider residual-based a posteriori error estimation for lowest-order nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. The resulting error estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. The error analysis is also shown to be applied to nonconforming finite element methods with face penalty and subgrid viscosity. Numerical results show that the estimator can be used to construct adaptive meshes.
    Print ISSN: 0008-0624
    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 12
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    Three-stage Hermite–Birkhoff solver of order 8 and 9 with variable step size for stiff ODEs (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-07-23
    Description: Variable-step (VS) \(3\) -stage Hermite–Birkhoff (HB) methods HB \((p=k+1)\) of order \(p=8\) and 9 are constructed as a combination of linear \(k\) -step methods of order \((p-2)\) and a diagonally implicit one-step \(3\) -stage Runge–Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge–Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A( \(\alpha \) )-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O \((p^2)\) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB( \(p\) ) of order \(p=8\) and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7–8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 13
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    A note on solving the fourth-order parabolic equation by the sinc-Galerkin method (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-06-17
    Description: A variable coefficient fourth-order partial differential equation in one space variable that arises in the study of transverse vibrations of a uniform flexible beam is examined. The numerical solution is obtained by using the sinc-Galerkin method. The method is then tested on three problems which have appeared in physical applications. The numerical results demonstrate the reliability and efficiency of the algorithm developed.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 14
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    The modified Newton–Shamanskii method for the solution of a quadratic vector equation arising in Markovian binary trees (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-05-28
    Description: In order to determine the extinction probability of a Markovian binary tree, it is necessary to find the minimal nonnegative solution of a quadratic vector equation. We apply the modified Newton–Shamanskii method for solving the equation. We show that the sequence of vectors generated by the modified Newton–Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 15
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    A mixed formulation for the direct approximation of the control of minimal $L^2$ (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-05-23
    Description: This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [ Cîndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013 ], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 16
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    Substructuring preconditioners for an $h$ - (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-05-13
    Description: We propose and study an iterative substructuring method for an $h$ - $p$ Nitsche-type discretization, following the original approach introduced in Bramble et al. Math. Comp. 47(175):103–134, ( 1986 ) for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments assess the performance of the preconditioner and verify the theory.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 17
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    Explicit multi-frequency symmetric extended RKN integrators for solving multi-frequency and multidimensional oscillatory reversible systems (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-04-23
    Description: This paper studies explicit multi-frequency symmetric extended Runge–Kutta–Nyström (ERKN) integrators tailored to numerically computing the multi-frequency and multidimensional oscillatory reversible second-order differential equations $q''(t)+Mq(t)=f\big (q(t)\big )$ . We establish the symmetry conditions in a simplified way for multi-frequency ERKN integrators. Five explicit multi-frequency symmetric ERKN integrators are derived based on the simplified symmetry conditions. The arbitrary high-order explicit multi-frequency symmetric ERKN integrators can be achieved by the application of the symmetric composition. The stability and phase properties of the new integrators are discussed. Five numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new explicit multi-frequency symmetric integrators when applied to the multi-frequency and multidimensional oscillatory reversible second-order differential equations.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 18
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    An adaptive sizing BFGS method for unconstrained optimization (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-04-23
    Description: In this paper, an adaptive sizing BFGS method for unconstrained optimization is proposed, whose scaling factor is automatically chosen by the signature of some terms of the approximate model’s curvature. The scaling factor is always chosen less than or equal to one as required by the convergence property of the self-scaling BFGS method of Al-Baali (Comput Optim Appl 9:191–203, 1998 ), while the choosing strategy can ensure the sufficient positive definiteness of the updating matrices. Under mild conditions, the global convergence properties of Al-Baali on convex functions are proved. Numerical results on some test problems show the proposed method is competitive with its counterparts provided that the scaling factor is not too small.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 19
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    Lavrentiev-type regularization methods for Hermitian problems (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-04-04
    Description: Lavrentiev regularization is a popular approach to the solution of linear discrete ill-posed problems with a Hermitian positive semidefinite matrix. This paper describes Lavrentiev-type regularization methods that can be applied to the solution of linear discrete ill-posed problems with a general Hermitian matrix. Fractional Lavrentiev-type methods as well as modifications suggested by the solution of certain matrix nearness problems are described. Computed examples illustrate the competitiveness of modified fractional Lavrentiev-type methods.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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  • 20
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    A solution procedure for constrained eigenvalue problems and its application within the structural finite-element code NOSA-ITACA (2014)
    Springer
    In: CALCOLO
    Details
    Publication Date: 2014-03-30
    Description: The paper presents an efficient and reliable implementation of numerical methods for constrained generalized eigenvalue problems, specialized for the modal analysis of linear elastic structures in a finite-element setting. The implementation, which takes into account the sparsity of the stiffness and mass matrices and the features of master-slave constraints, is based on open-source packages embedded in the finite-element code NOSA-ITACA. Numerical tests on historical building are performed, with the aims of calculating their vibration frequencies and mode shape vectors, comparing them to the results of a general purpose commercial code and assessing the accuracy of the tool developed.
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    Electronic ISSN: 1126-5434
    Topics: Mathematics
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