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  • Artikel  (2.369)
  • Oxford University Press  (2.369)
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  • 1
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    Localized pointwise error estimates for direct flux approximation (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: Pointwise error estimates for the first-order div least-squares (LS) finite element method for second-order elliptic partial differential equations are presented. Direct flux approximation is considered as an important advantage of the LS method. However, there are no known pointwise error estimates for the direct flux approximation. In this paper, we provide optimal pointwise estimates which show local dependence of the error at a point and weak dependence of the global norm. As an elementary consequence of these estimates, we provide an asymptotic error expansion inequality. The inequality has applications to superconvergence and a posteriori estimates.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 2
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    Fast multiscale regularization methods for high-order numerical differentiation (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: The first-order and higher-order derivatives of a function can be viewed as the solutions of Volterra integral equations of the first kind. In this paper we propose a fast multiscale solver for the numerical solution of the Tikhonov regularization of the Volterra equations. In association with the special form of the kernels, the matrices resulting from the discretization by multiscale bases are sparse. Moreover, they can be truncated using proper strategies with only a minor loss of accuracy. In the best case, the number of nonzero entries of the truncated matrices is linear with respect to the dimensions of the matrices. The accuracy of the solution from the solver is analysed theoretically and verified by numerical experiments.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 3
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    Convergence and supercloseness of a finite element method for a singularly perturbed convection-diffusion problem on an L-shaped domain (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: The finite element method with $\mathscr {Q}_p$ elements is applied to a singularly perturbed convection–diffusion problem on an L-shaped domain. As an effect of corner singularities the exact solution is not $H^2$ -regular. Therefore, we combine a layer-adapted Shishkin mesh with a special grading adapted to the corner singularity. On such meshes we prove error estimates and estimates for the closeness error which explicitly show the influence of the grading parameter $\mu$ . Hence, $\mu$ can be chosen such that optimal error bounds are obtained. Thereby, it turns out that in the problem studied the influence of the corner singularity becomes small if the perturbation parameter $\varepsilon$ decreases. Moreover, we conduct numerical experiments that verify the theoretical results.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 4
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    Ptak's nondiscrete induction and its application to matrix iterations (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: Pták's method of nondiscrete induction is based on the idea that in the analysis of iterative processes one should aim at rates of convergence as functions rather than just numbers, because functions may give convergence estimates that are tight throughout the iteration rather than just asymptotically. In this paper we motivate and prove a theorem on nondiscrete induction, originally due to Potra and Pták, and we apply it to the Newton iterations for computing the matrix polar decomposition and the matrix square root. Our goal is to illustrate the application of the method of nondiscrete induction in the finite-dimensional numerical linear algebra context. We show the sharpness of the resulting convergence estimate analytically for the polar decomposition iteration and on some examples for the square root iteration. We also discuss some of the method's limitations and possible extensions.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 5
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    A first-order system Petrov-Galerkin discretization for a reaction-diffusion problem on a fitted mesh (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: We consider the numerical solution, by a Petrov–Galerkin finite-element method, of a singularly perturbed reaction–diffusion differential equation posed on the unit square. In Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. , 50 , 2729–2743), it is argued that the natural energy norm, associated with a standard Galerkin approach, is not an appropriate setting for analysing such problems, and there they propose a method for which the natural norm is ‘balanced’. In the style of a first-order system least squares method, we extend the approach of Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. , 50 , 2729–2743) by introducing a constraint which simplifies the associated finite-element space and the method's analysis. We prove robust convergence in a balanced norm on a piecewise-uniform (Shishkin) mesh, and present supporting numerical results. Finally, we demonstrate how the resulting linear systems are solved optimally using multigrid methods.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 6
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    Tensor-product surface patches with Pythagorean-hodograph isoparametric curves (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: The construction of tensor-product surface patches with a family of Pythagorean-hodograph (PH) isoparametric curves is investigated. The simplest nontrivial instances, interpolating four prescribed patch boundary curves, involve degree $(5,4)$ tensor-product surface patches $\bf{x}(u,v)$ whose $v=\hbox {constant}$ isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closed-form solution for this type of equation is derived, and conditions for the existence of solutions are identified. The surfaces incorporate three residual scalar freedoms which can be exploited to improve the interior shape of the patch. The implementation of the method is illustrated through a selection of computed examples.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 7
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    Computing interior eigenvalues of domains from far fields (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: Interior eigenvalues of bounded scattering objects can be rigorously characterized from multi-static and multi-frequency far field data, that is, from the behaviour of scattered waves far away from the object. This characterization, the so-called inside–outside duality, holds for various types of penetrable and impenetrable scatterers and is based on the behaviour of a particular eigenvalue of the far field operator. It naturally leads to a numerical algorithm for computing interior eigenvalues of a scatterer that does not require shape or physical properties of the scatterer as input. Since the nonlinear inverse problem to compute such interior eigenvalues from far field data is ill-posed, we propose a regularizing algorithm that is shown to converge as the noise level of the far field data tends to zero. We illustrate feasibility and accuracy of our algorithm by numerical experiments where we compute interior transmission eigenvalues and Robin eigenvalues of the Laplacian in three-dimensional domains from scattering data of these domains due to plane incident waves.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 8
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    A posteriori error estimates for fully discrete fractional-step {vartheta}-approximations for parabolic equations (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: We derive optimal-order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretization in time we apply the fractional-step $\vartheta $ -scheme, and for the discretization in space the finite element method with finite element spaces that are allowed to change with time. The first optimal-order a posteriori error estimates for the norms of $L^\infty (0,T;L^2(\varOmega ))$ and $L^2(0,T;H^1(\varOmega ))$ are derived by applying the reconstruction technique.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 9
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    A fast and accurate numerical method for symmetric Levy processes based on the Fourier transform and sinc-Gauss sampling formula (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: In this paper, we propose a fast and accurate numerical method based on Fourier transforms to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transforms proposed by Ooura. These formulas are combined with nonuniform fast Fourier transforms (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a nonsmooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appear to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.
    Print ISSN: 0272-4979
    Digitale ISSN: 1464-3642
    Thema: Mathematik
    Publiziert von Oxford University Press
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  • 10
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    Spectral discretization of Darcy's equations coupled with the heat equation (2016)
    Oxford University Press
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    Publikationsdatum: 2016-07-13
    Beschreibung: In this paper, we consider the heat equation coupled with Darcy's law with a nonlinear source term describing heat production due to an exothermic chemical reaction. The existence and uniqueness of a solution are established. Next, a spectral discretization of the problem is presented and thoroughly analysed. Finally, we present some numerical experiments which confirm the interest of the discretization.
    Print ISSN: 0272-4979
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    Thema: Mathematik
    Publiziert von Oxford University Press
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