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:M. Esmaeilbeigi; O. Chatrabgoun; M. Shafa Abstract: Abstract In many practical problems, it is often desirable to interpolate not only the function values but also the values of derivatives up to certain order, as in the Hermite interpolation. The Hermite interpolation method by radial basis functions is used widely for solving scattered Hermite data approximation problems. However, sometimes it makes more sense to approximate the solution by a least squares fit. This is particularly true when the data are contaminated with noise. In this paper, a weighted meshless method is presented to solve least squares problems with noise. The weighted meshless method by Gaussian radial basis functions is proposed to fit scattered Hermite data with noise in certain local regions of the problem’s domain. Existence and uniqueness of the solution is proved. This approach has one parameter which can adjust the accuracy according to the size of the noise. Another advantage of the weighted meshless method is that it can be used for problems in high dimensions with nonregular domains. The numerical experiments show that our weighted meshless method has better performance than the traditional least squares method in the case of noisy Hermite data. PubDate: 2017-09-06 DOI: 10.1007/s10444-017-9555-7

Authors:Martin Halla; Lothar Nannen Abstract: Abstract We consider the numerical solution of the Helmholtz equation in domains with one infinite cylindrical waveguide. Such problems exhibit wavenumbers on different scales in the vicinity of cut-off frequencies. This leads to performance issues for non-modal methods like the perfectly matched layer or the Hardy space infinite element method. To improve the latter, we propose a two scale Hardy space infinite element method which can be optimized for wavenumbers on two different scales. It is a tensor product Galerkin method and fits into existing analysis. Up to arbitrary small thresholds it converges exponentially with respect to the number of longitudinal unknowns in the waveguide. Numerical experiments support the theoretical error bounds. PubDate: 2017-09-06 DOI: 10.1007/s10444-017-9549-5

Authors:Woinshet D. Mergia; Kailash C. Patidar Abstract: Abstract We consider a predator-prey model arising in ecology that describes a slow-fast dynamical system. The dynamics of the model is expressed by a system of nonlinear differential equations having different time scales. Designing numerical methods for solving problems exhibiting multiple time scales within a system, such as those considered in this paper, has always been a challenging task. To solve such complicated systems, we therefore use an efficient time-stepping algorithm based on fractional-step methods. To develop our algorithm, we first decouple the original system into fast and slow sub-systems, and then apply suitable sub-algorithms based on a class of θ-methods, to discretize each sub-system independently using different time-steps. Then the algorithm for the full problem is obtained by utilizing a higher-order product method by merging the sub-algorithms at each time-step. The nonlinear system resulting from the use of implicit schemes is solved by two different nonlinear solvers, namely, the Jacobian-free Newton-Krylov method and the well-known Anderson’s acceleration technique. The fractional-step θ-methods give us flexibility to use a variety of methods for each sub-system and they are able to preserve qualitative properties of the solution. We analyze these methods for stability and convergence. Several numerical results indicating the efficiency of the proposed method are presented. We also provide numerical results that confirm our theoretical investigations. PubDate: 2017-08-24 DOI: 10.1007/s10444-017-9554-8

Authors:Linghua Chen; Espen Robstad Jakobsen; Arvid Naess Abstract: Abstract We study a numerical method to compute probability density functions of solutions of stochastic differential equations. The method is sometimes called the numerical path integration method and has been shown to be fast and accurate in application oriented fields. In this paper we provide a rigorous analysis of the method that covers systems of equations with unbounded coefficients. Working in a natural space for densities, L 1, we obtain stability, consistency, and new convergence results for the method, new well-posedness and semigroup generation results for the related Fokker-Planck-Kolmogorov equation, and a new and rigorous connection to the corresponding probability density functions for both the approximate and the exact problems. To prove the results we combine semigroup and PDE arguments in a new way that should be of independent interest. PubDate: 2017-08-19 DOI: 10.1007/s10444-017-9558-4

Authors:Zhiping Mao; Jie Shen Abstract: Abstract Solutions of two-sided fractional differential equations (FDEs) usually exhibit singularities at the both endpoints, so it can not be well approximated by a usual polynomial based method. Furthermore, the singular behaviors are usually not known a priori, making it difficult to construct special spectral methods tailored for given singularities. We construct a spectral element approximation with geometric mesh, describe its efficient implementation, and derive corresponding error estimates. We also present ample numerical examples to validate our error analysis. PubDate: 2017-08-18 DOI: 10.1007/s10444-017-9561-9

Authors:Tao Sun; Penghang Yin; Lizhi Cheng; Hao Jiang Abstract: Abstract In this paper, we consider the minimization of a class of nonconvex composite functions with difference of convex structure under linear constraints. While this kind of problems in theory can be solved by the celebrated alternating direction method of multipliers (ADMM), a direct application of ADMM often leads to difficult nonconvex subproblems. To address this issue, we propose to convexify the subproblems through a linearization technique as done in the difference of convex functions algorithm (DCA). By assuming the Kurdyka-Łojasiewicz property, we prove that the resulting algorithm sequentially converges to a critical point. It turns out that in the applications of signal and image processing such as compressed sensing and image denoising, the proposed algorithm usually enjoys closed-form solutions of the subproblems and thus can be very efficient. We provide numerical experiments to demonstrate the effectiveness of our algorithm. PubDate: 2017-08-16 DOI: 10.1007/s10444-017-9559-3

Authors:Sudeep Kundu; Amiya Kumar Pani Abstract: Abstract In this article, we discuss global stabilization results for the Burgers’ equation using nonlinear Neumann boundary feedback control law. As a result of the nonlinear feedback control, a typical nonlinear problem is derived. Then, based on C 0-conforming finite element method, global stabilization results for the semidiscrete solution are analyzed. Further, introducing an auxiliary projection, optimal error estimates in \(L^{\infty }(L^{2})\) , \(L^{\infty }(H^{1})\) and \(L^{\infty }(L^{\infty })\) -norms for the state variable are obtained. Moreover, superconvergence results are established for the first time for the feedback control laws, which preserve exponential stabilization property. Finally, some numerical experiments are conducted to confirm our theoretical findings. PubDate: 2017-08-15 DOI: 10.1007/s10444-017-9553-9

Authors:Guo-Dong Zhang; Jinjin Yang; Chunjia Bi Abstract: Abstract In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L 2 and H 1 fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems. PubDate: 2017-08-15 DOI: 10.1007/s10444-017-9552-x

Authors:Stéphane Clain; Raphaël Loubère; Gaspar J. Machado Abstract: Abstract We propose a new family of high order accurate finite volume schemes devoted to solve one-dimensional steady-state hyperbolic systems. High-accuracy (up to the sixth-order presently) is achieved thanks to polynomial reconstructions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of discontinuities. Such a procedure demands the determination of a detector chain to discriminate between troubled and valid cells, a cascade of polynomial degrees to be successively tested when oscillations are detected, and a parachute scheme corresponding to the last, viscous, and robust scheme of the cascade. Experimented on linear, Burgers’, and Euler equations, we demonstrate that the schemes manage to retrieve smooth solutions with optimal order of accuracy but also irregular solutions without spurious oscillations. PubDate: 2017-08-12 DOI: 10.1007/s10444-017-9556-6

Authors:Tingchun Wang; Xiaofei Zhao; Jiaping Jiang Abstract: Abstract The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H 2-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h 2 + τ 2) with mesh-size h and time step τ in the discrete H 2-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes. PubDate: 2017-08-10 DOI: 10.1007/s10444-017-9557-5

Authors:Pedro R. S. Antunes Abstract: Abstract The method of fundamental solutions (MFS) is a meshless method for solving boundary value problems with some partial differential equations. It allows to obtain highly accurate approximations for the solutions assuming that they are smooth enough, even with small matrices. As a counterpart, the (dense) matrices involved are often ill-conditioned which is related to the well known uncertainty principle stating that it is impossible to have high accuracy and good conditioning at the same time. In this work, we propose a technique to reduce the ill conditioning in the MFS, assuming that the source points are placed on a circumference of radius R. The idea is to apply a suitable change of basis that provides new basis functions that span the same space as the MFS’s, but are much better conditioned. In the particular case of circular domains, the algorithm allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of points sources and R. PubDate: 2017-08-05 DOI: 10.1007/s10444-017-9548-6

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