Authors:Dianming Hou; Chuanju Xu Pages: 911 - 944 Abstract: In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x 1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods. PubDate: 2017-10-01 DOI: 10.1007/s10444-016-9511-y Issue No:Vol. 43, No. 5 (2017)

Authors:Yanzhao Cao; Ying Jiang; Yuesheng Xu Pages: 973 - 998 Abstract: The goal of this paper is to construct an efficient numerical algorithm for computing the coefficient matrix and the right hand side of the linear system resulting from the spectral Galerkin approximation of a stochastic elliptic partial differential equation. We establish that the proposed algorithm achieves an exponential convergence with requiring only O \((n\log _{2}^{d+1}n)\) number of arithmetic operations, where n is the highest degree of the one dimensional orthogonal polynomial used in the algorithm, d+1 is the number of terms in the finite Karhunen–Loéve (K-L) expansion. Numerical experiments confirm the theoretical estimates of the proposed algorithm and demonstrate its computational efficiency. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9513-4 Issue No:Vol. 43, No. 5 (2017)

Authors:Zhanjing Tao; Jianxian Qiu Pages: 1023 - 1058 Abstract: In this paper, a class of high-order central Hermite WENO (HWENO) schemes based on finite volume framework and staggered meshes is proposed for directly solving one- and two-dimensional Hamilton-Jacobi (HJ) equations. The methods involve the Lax-Wendroff type discretizations or the natural continuous extension of Runge-Kutta methods in time. This work can be regarded as an extension of central HWENO schemes for hyperbolic conservation laws (Tao et al. J. Comput. Phys. 318, 222–251, 2016) which combine the central scheme and the HWENO spatial reconstructions and therefore carry many features of both schemes. Generally, it is not straightforward to design a finite volume scheme to directly solve HJ equations and a key ingredient for directly solving such equations is the reconstruction of numerical Hamiltonians to guarantee the stability of methods. Benefited from the central strategy, our methods require no numerical Hamiltonians. Meanwhile, the zeroth-order and the first-order moments of the solution are involved in the spatial HWENO reconstructions which is more compact compared with WENO schemes. The reconstructions are implemented through a dimension-by-dimension strategy when the spatial dimension is higher than one. A collection of one- and two- dimensional numerical examples is performed to validate high resolution and robustness of the methods in approximating the solutions of HJ equations, which involve linear, nonlinear, smooth, non-smooth, convex or non-convex Hamiltonians. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9515-2 Issue No:Vol. 43, No. 5 (2017)

Authors:Caroline Moosmüller Pages: 1059 - 1074 Abstract: We propose a new adaption of linear Hermite subdivision schemes to the manifold setting. Our construction is intrinsic, as it is based solely on geodesics and on the parallel transport operator of the manifold. The resulting nonlinear Hermite subdivision schemes are analyzed with respect to convergence and C 1 smoothness. Similar to previous work on manifold-valued subdivision, this analysis is carried out by proving that a so-called proximity condition is fulfilled. This condition allows to conclude convergence and smoothness properties of the manifold-valued scheme from its linear counterpart, provided that the input data are dense enough. Therefore the main part of this paper is concerned with showing that our nonlinear Hermite scheme is “close enough”, i.e., in proximity, to the linear scheme it is derived from. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9516-1 Issue No:Vol. 43, No. 5 (2017)

Authors:Wenqing Wang; Xuehai Huang; Kai Tang; Ruiyue Zhou Abstract: Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates. PubDate: 2017-11-24 DOI: 10.1007/s10444-017-9572-6

Authors:Annie Cuyt; Wen-shin Lee Abstract: The problem of multivariate exponential analysis or sparse interpolation has received a lot of attention, especially with respect to the number of samples required to solve it unambiguously. In this paper we show how to bring the number of samples down to the absolute minimum of (d + 1)n where d is the dimension of the problem and n is the number of exponential terms. To this end we present a fundamentally different approach for the multivariate problem statement. We combine a one-dimensional exponential analysis method such as ESPRIT, MUSIC, the matrix pencil or any Prony-like method, with some linear systems of equations because the multivariate exponents are inner products and thus linear expressions in the parameters. PubDate: 2017-11-16 DOI: 10.1007/s10444-017-9570-8

Authors:Bo Wang; Li-Lian Wang; Ziqing Xie Abstract: We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map. PubDate: 2017-11-08 DOI: 10.1007/s10444-017-9569-1

Authors:Huadong Gao; Weiwei Sun Abstract: A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg-Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field σ = c u r l A as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional nonconvex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains. PubDate: 2017-11-07 DOI: 10.1007/s10444-017-9568-2

Authors:Min Tao; Xiaoming Yuan Abstract: Recently, the alternating direction method of multipliers (ADMM) has found many efficient applications in various areas; and it has been shown that the convergence is not guaranteed when it is directly extended to the multiple-block case of separable convex minimization problems where there are m ≥ 3 functions without coupled variables in the objective. This fact has given great impetus to investigate various conditions on both the model and the algorithm’s parameter that can ensure the convergence of the direct extension of ADMM (abbreviated as “e-ADMM”). Despite some results under very strong conditions (e.g., at least (m − 1) functions should be strongly convex) that are applicable to the generic case with a general m, some others concentrate on the special case of m = 3 under the relatively milder condition that only one function is assumed to be strongly convex. We focus on extending the convergence analysis from the case of m = 3 to the more general case of m ≥ 3. That is, we show the convergence of e-ADMM for the case of m ≥ 3 with the assumption of only (m − 2) functions being strongly convex; and establish its convergence rates in different scenarios such as the worst-case convergence rates measured by iteration complexity and the globally linear convergence rate under stronger assumptions. Thus the convergence of e-ADMM for the general case of m ≥ 4 is proved; this result seems to be still unknown even though it is intuitive given the known result of the case of m = 3. Even for the special case of m = 3, our convergence results turn out to be more general than the existing results that are derived specifically for the case of m = 3. PubDate: 2017-10-13 DOI: 10.1007/s10444-017-9560-x

Authors:Qi Hong; Jiming Wu Abstract: In this paper, we study a so-called modified Q 1-finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q 1-finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h 1+γ -parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H 1 error estimate of the modified Q 1-finite volume element scheme is obtained without the traditional h 1+γ -parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis. PubDate: 2017-10-12 DOI: 10.1007/s10444-017-9567-3

Authors:Hong Lu; Peter W. Bates; Wenping Chen; Mingji Zhang Abstract: We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (−△) α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth. PubDate: 2017-10-09 DOI: 10.1007/s10444-017-9564-6

Authors:Stéphane Clain; Raphaël Loubère; Gaspar J. Machado Abstract: During typesetting, Figs. 8 and 21 got corrupted and the images shown in the online published version are not correct. The original publication was updated. PubDate: 2017-10-07 DOI: 10.1007/s10444-017-9563-7

Authors:L. Iapichino; S. Ulbrich; S. Volkwein Abstract: In this paper the reduced basis (RB) method is applied to solve quadratic multiobjective optimal control problems governed by linear parametrized variational equations. These problems often arise in applications, where the quality of the system behavior has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued linear-quadratic optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the RB method. The accuracy is guaranteed by an a-posteriori error estimate. An effective sensitivity analysis allows to further reduce the computational times for identifying a suitable and representative set of optimal controls. PubDate: 2017-10-01 DOI: 10.1007/s10444-016-9512-x

Authors:Natalia Kopteva; Torsten Linß Abstract: Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backward Euler and of the Crank-Nicolson methods are considered. For their full discretizations, we use elliptic reconstructions that are, respectively, piecewise-constant and piecewise-linear in time. Certain bounds for the Green’s function of the parabolic operator are also employed. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9514-3

Authors:Ghulam Mustafa; Rabia Hameed Abstract: Families of parameter dependent univariate and bivariate subdivision schemes are presented in this paper. These families are new variants of the Lane-Riesenfeld algorithm. So the subdivision algorithms consist of both refining and smoothing steps. In refining step, we use the quartic B-spline based subdivision schemes. In smoothing step, we average the adjacent points. The bivariate schemes are the non-tensor product version of our univariate schemes. Moreover, for odd and even number of smoothing steps, we get the primal and dual schemes respectively. Higher regularity of the schemes can be achieved by increasing the number of smoothing steps. These schemes can be nicely generalized to contain local shape parameters that allow the user to adjust locally the shape of the limit curve/surface. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9519-y

Authors:Dao Huy Cuong; Mai Duc Thanh Abstract: A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The construction of our scheme is based on exact solutions in computational form of local Riemann problems. Numerical tests are conducted, where comparisons between this van Leer-type scheme and a Godunov-type scheme are provided. Data for the tests are taken in both the subcritical region as well as supercritical region. Especially, tests for resonant cases where the exact solutions contain coinciding waves are also investigated. All numerical tests show that each of these two methods can give a good accuracy, while the van Leer -type scheme gives a better accuracy than the Godunov-type scheme. Furthermore, it is shown that the van Leer-type scheme is also well-balanced in the sense that it can capture exactly stationary contact discontinuity waves. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9521-4

Authors:X. Claeys; R. Hiptmair; E. Spindler Abstract: We consider isotropic scalar diffusion boundary value problems whose diffusion coefficients are piecewise constant with respect to a partition of space into Lipschitz subdomains. We allow so-called material junctions where three or more subdomains may abut. We derive a boundary integral equation of the second kind posed on the skeleton of the subdomain partition that involves, as unknown, only one trace function at each point of each interface. We prove the well-posedness of the corresponding boundary integral equations. We also report numerical tests for Galerkin boundary element discretisations, in which the new approach proves to be highly competitive compared to the well-established first kind direct single-trace boundary integral formulation. In particular, GMRES seems to enjoy fast convergence independent of the mesh resolution for the discrete second kind BIE. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9517-0

Authors:A. F. Hegarty; E. O’Riordan Abstract: A linear singularly perturbed elliptic problem, of convection-diffusion type, posed on a circular domain is examined. Regularity constraints are imposed on the data in the vicinity of the two characteristic points. The solution is decomposed into a regular and a singular component. A priori parameter-explicit pointwise bounds on the partial derivatives of these components are established. By transforming to polar co-ordinates, a monotone finite difference method is constructed on a piecewise-uniform layer-adapted mesh of Shishkin type. Numerical analysis is presented for this monotone numerical method. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established. PubDate: 2017-10-01 DOI: 10.1007/s10444-016-9510-z

Authors:J. A. Ezquerro; M. A. Hernández-Verón Abstract: This paper focuses on the importance of center conditions on the first derivative of the operator involved in the solution of nonlinear equations by Newton’s method when the semilocal convergence of the method is established from the technique of recurrence relations. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9518-z

Authors:Theresa Wenger; Sina Ober-Blöbaum; Sigrid Leyendecker Abstract: In this work, variational integrators of higher order for dynamical systems with holonomic constraints are constructed and analyzed. The construction is based on approximating the configuration and the Lagrange multiplier via different polynomials. The splitting of the augmented Lagrangian in two parts enables the use of different quadrature formulas to approximate the integral of each part. Conditions are derived that ensure the linear independence of the higher order constrained discrete Euler-Lagrange equations and stiff accuracy. Time reversibility is investigated for the discrete flow on configuration level only as for the flow on configuration and momentum level. The fulfillment of the hidden constraints plays an important role for the time reversibility of the presented integrators. The order of convergence is investigated numerically. Order reduction of the momentum and the Lagrange multiplier compared to the order of the configuration occurs in general, but can be avoided by fulfilling the hidden constraints in a simple post processing step. Regarding efficiency versus accuracy a numerical analysis yields that higher orders increase the accuracy of the discrete solution substantially while the computational costs decrease. A comparison to the constrained Galerkin methods in Marsden and West (Acta Numerica 10, 357–514 2001) and the symplectic SPARK integrators of Jay (SIAM Journal on Numerical Analysis 45(5), 1814–1842 2007) reveals that the approach presented here is more general and thus allows for more flexibility in the design of the integrator. PubDate: 2017-10-01 DOI: 10.1007/s10444-017-9520-5