Authors:Adrian Holhoş; Daniela Roşca Pages: 677 - 697 Abstract: Abstract We construct a bijective continuous area preserving map from a class of elongated dipyramids to the sphere, together with its inverse. Then we investigate for which such solid polyhedrons the area preserving map can be used for constructing a bijective continuous volume preserving map to the 3D-ball. These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and refinable grids on the elongated dipyramids. In particular, we show that HEALPix grids can be obtained from these maps. We also study the optimality of the logarithmic energy of the configurations of points obtained from these grids. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9502-z Issue No:Vol. 43, No. 4 (2017)

Authors:Zhijiang Zhang; Weihua Deng Pages: 699 - 732 Abstract: Abstract The functional distributions of particle trajectories have wide applications. This paper focuses on providing effective computation methods for the models, which characterize the distribution of the functionals of the paths of anomalous diffusion with both traps and flights. Two kinds of discretization schemes are proposed for the time fractional substantial derivatives. The Galerkin method with interval spline scaling bases is used for the space approximation; compared with the usual finite element or spectral polynomial bases, the spline scaling bases have the advantages of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The rigorous stability analyses for both the semi and the full discrete schemes are skillfully developed. Under the assumptions of the regularity of the exact solution, the convergence of the provided schemes is also theoretically proved and numerically verified. Moreover, the theoretical background of the selected basis function and the implementation details of the algorithms involved are described in detail. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9503-y Issue No:Vol. 43, No. 4 (2017)

Authors:Amir Averbuch; Pekka Neittaanmäki; Etay Shefi; Valery Zheludev Pages: 733 - 758 Abstract: Abstract In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are designed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, the splines use either clean or noised arbitrarily-spaced samples. Formulas for the spline’s extrapolation beyond the sampling interval are established. Sharp estimations of the approximation errors are presented. The capability to adapt the grid to the structure of an object and to have minimal requirements to the operating memory are of great advantages for offline processing of signals and multidimensional data arrays. The designed splines serve as a source for generating real-time wavelet transforms to apply to signals in scenarios where the signal’s samples subsequently arrive one after the other at random times. The wavelet transforms are executed by six-tap weighted moving averages of the signal’s samples without delay. On arrival of new samples, only a couple of adjacent transform coefficients are updated in a way that no boundary effects arise. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9504-x Issue No:Vol. 43, No. 4 (2017)

Authors:Jin Zhang; Xiaowei Liu Pages: 759 - 775 Abstract: Abstract In this paper, we analyze the supercloseness property of the streamline diffusion finite element method (SDFEM) on Shishkin triangular meshes, which is different from one in the case of rectangular meshes. The analysis depends on integral inequalities for the parts related to the diffusion in the bilinear form. Moreover, our result allows the construction of a simple postprocessing that yields a more accurate solution. Finally, numerical experiments support these theoretical results. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9505-9 Issue No:Vol. 43, No. 4 (2017)

Authors:Stanisław Lewanowicz; Paweł Keller; Paweł Woźny Pages: 777 - 793 Abstract: Abstract Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L 2-solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9506-8 Issue No:Vol. 43, No. 4 (2017)

Authors:Waixiang Cao; Xu Zhang; Zhimin Zhang Pages: 795 - 821 Abstract: Abstract In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9507-7 Issue No:Vol. 43, No. 4 (2017)

Authors:Yao Rong; Yanren Hou; Yuhong Zhang Pages: 823 - 848 Abstract: Abstract In this paper, we construct a second order algorithm based on the spectral deferred correction method for the time-dependent magnetohydrodynamics flows at a low magnetic Reynolds number. We present a complete theoretical analysis to prove that this algorithm is unconditionally stable, consistent and second order accuracy. Finally, two numerical examples are given to illustrate the convergence and effectiveness of our algorithm. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9508-6 Issue No:Vol. 43, No. 4 (2017)

Authors:Rosa Donat; Sergio López-Ureña; Maria Santágueda Pages: 849 - 883 Abstract: Abstract In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Power p schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Power p schemes and it is based on a weighted analog of the Power p mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Power p schemes. PubDate: 2017-08-01 DOI: 10.1007/s10444-016-9509-5 Issue No:Vol. 43, No. 4 (2017)

Authors:Young Ok Choi; Johannes Tausch Pages: 473 - 493 Abstract: Abstract Since the fundamental solution for transient Stokes flow in three dimensions is complicated it is difficult to implement discretization methods for boundary integral formulations. We derive a representation of the Stokeslet and stresslet in terms of incomplete gamma functions and investigate the nature of the singularity of the single- and double layer potentials. Further, we give analytical formulas for the time integration and develop Galerkin schemes with tensor product piecewise polynomial ansatz functions. Numerical results demonstrate optimal convergence rates. PubDate: 2017-06-01 DOI: 10.1007/s10444-016-9493-9 Issue No:Vol. 43, No. 3 (2017)

Authors:Jintao Cui; Thirupathi Gudi Pages: 517 - 535 Abstract: Abstract In this article, we study a finite element approximation for a model free boundary plasma problem. Using a mixed approach (which resembles an optimal control problem with control constraints), we formulate a weak formulation and study the existence and uniqueness of a solution to the continuous model problem. Using the same setting, we formulate and analyze the discrete problem. We derive optimal order energy norm a priori error estimates proving the convergence of the method. Further, we derive a reliable and efficient a posteriori error estimator for the adaptive mesh refinement algorithm. Finally, we illustrate the theoretical results by some numerical examples. PubDate: 2017-06-01 DOI: 10.1007/s10444-016-9495-7 Issue No:Vol. 43, No. 3 (2017)

Authors:Erich Novak; Mario Ullrich; Henryk Woźniakowski; Shun Zhang Pages: 537 - 553 Abstract: Abstract We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space \(H^{s} (\mathbb {R})\) and from the space \(C^{s}(\mathbb {R})\) with an arbitrary integer s ≥ 1. We find tight upper and lower bounds for the worst case error of optimal algorithms that use n function values. More specifically, we study integrals of the form 1 $$ I_{k}^{\varrho} (f) = {\int}_{\mathbb{R}} f(x) \,\mathrm{e}^{-i\,kx} \varrho(x) \, \mathrm{d} x\ \ \ \text{for}\ \ f\in H^{s}(\mathbb{R})\ \ \text{or}\ \ f\in C^{s}(\mathbb{R}) $$ with \(k\in {\mathbb {R}}\) and a smooth density function ρ such as \( \rho (x) = \frac {1}{\sqrt {2 \pi }} \exp (-x^{2}/2)\) . The optimal error bounds are \({\Theta }((n+\max (1, k ))^{-s})\) with the factors in the Θ notation dependent only on s and ϱ. PubDate: 2017-06-01 DOI: 10.1007/s10444-016-9496-6 Issue No:Vol. 43, No. 3 (2017)

Authors:Vahagn Bayramyan; Hakop Hakopian Pages: 607 - 626 Abstract: Abstract In this paper we consider n-poised planar node sets, as well as more special ones, called G C n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ℓ is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \) meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n − 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C n sets. At the end we present a conjecture concerning any k-node line. PubDate: 2017-06-01 DOI: 10.1007/s10444-016-9499-3 Issue No:Vol. 43, No. 3 (2017)

Authors:Roman Chapko; Drossos Gintides; Leonidas Mindrinos Abstract: Abstract In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method. PubDate: 2017-07-31 DOI: 10.1007/s10444-017-9550-z

Authors:Alina Chertock; Yekaterina Epshteyn; Hengrui Hu; Alexander Kurganov Abstract: Abstract Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis phenomenon plays an important role in bacteria/cell aggregation and pattern formation mechanisms, as well as in tumor growth. A common property of all chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in rapid growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. There is consequently a need for accurate and computationally efficient numerical methods for the chemotaxis models. In this work, we develop and study novel high-order hybrid finite-volume-finite-difference schemes for the Patlak-Keller-Segel chemotaxis system and related models. We demonstrate high-accuracy, stability and computational efficiency of the proposed schemes in a number of numerical examples. PubDate: 2017-07-21 DOI: 10.1007/s10444-017-9545-9

Authors:Chao Zeng; Jiansong Deng Abstract: Abstract T-meshes are a type of rectangular partitions of planar domains which allow hanging vertices. Because of the special structure of T-meshes, adaptive local refinement is possible for splines defined on this type of meshes, which provides a solution for the defect of NURBS. In this paper, we generalize the definitions to the three-dimensional (3D) case and discuss a fundamental problem – the dimension of trivariate spline spaces on 3D T-meshes. We focus on a special case where splines are C d−1 continuous for degree d. The smoothing cofactor method for trivariate splines is explored for this situation. We obtain a general dimension formula and present lower and upper bounds for the dimension. At last, we introduce a type of 3D T-meshes, where we can give an explicit dimension formula. PubDate: 2017-07-12 DOI: 10.1007/s10444-017-9551-y

Authors:Alberto Gil C. P. Ramos Abstract: Abstract The current paper concerns the uniform and high-order discretization of the novel approach to the computation of Sturm–Liouville problems via Fer streamers, put forth in Ramos and Iserles (Numer. Math. 131(3), 541—565 2015). In particular, the discretization schemes are shown to enjoy large step sizes uniform over the entire eigenvalue range and tight error estimates uniform for every eigenvalue. They are made explicit for global orders 4,7,10. In addition, the present paper provides total error estimates that quantify the interplay between the truncation and the discretization in the approach by Fer streamers. PubDate: 2017-07-04 DOI: 10.1007/s10444-017-9547-7

Authors:Haitao Leng; Yanping Chen Abstract: Abstract In this paper we study the convergence of an adaptive finite element method for optimal control problems with integral control constraint. For discretization, we use piecewise constant discretization for the control and continuous piecewise linear discretization for the state and the co-state. The contraction, between two consecutive loops, is proved. Additionally, we find the adaptive finite element method has the optimal convergence rate. In the end, we give some examples to support our theoretical analysis. PubDate: 2017-07-03 DOI: 10.1007/s10444-017-9546-8

Authors:Aziz Takhirov; Alexander Lozovskiy Abstract: Abstract The nonlinear filter based stabilization proposed in Layton et al. (J. Math. Fluid Mech. 14(2), 325–354 2012) allows to incorporate an eddy viscosity model into an existing laminar flow codes in a modular way. However, the proposed nonlinear filtering step requires the assembly of the associated matrix at each time step and solving a linear system with an indefinte matrix. We propose computationally efficient version of the filtering step that only requires the assembly once, and the solution of two symmetric, positive definite systems at each time step. We also test a new indicator function based on the entropy viscosity model of Guermond (Int. J. Numer. Meth. Fluids. 57(9), 1153–1170 2008); Guermond et al. (J. Sci. Comput. 49(1), 35–50 2011). PubDate: 2017-06-21 DOI: 10.1007/s10444-017-9544-x

Authors:José L. López Abstract: Abstract We consider the Bessel functions J ν (z) and Y ν (z) for R ν > −1/2 and R z ≥ 0. We derive a convergent expansion of J ν (z) in terms of the derivatives of \((\sin z)/z\) , and a convergent expansion of Y ν (z) in terms of derivatives of \((1-\cos z)/z\) , derivatives of (1 − e −z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments. PubDate: 2017-06-19 DOI: 10.1007/s10444-017-9543-y

Authors:Qiya Hu; Long Yuan Abstract: Abstract In this paper we are concerned with plane wave discretizations of nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. To this end, we design a plane wave method combined with local spectral elements for the discretization of such nonhomogeneous equations. This method contains two steps: we first solve a series of nonhomogeneous local problems on auxiliary smooth subdomains by the spectral element method, and then apply the plane wave method to the discretization of the resulting (locally homogeneous) residue problem on the global solution domain. We derive error estimates of the approximate solutions generated by this method. The numerical results show that the resulting approximate solutions possess high accuracy. PubDate: 2017-06-09 DOI: 10.1007/s10444-017-9542-z