Authors:Matthew J. Colbrook; Athanasisos S. Fokas Pages: 1 - 17 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Matthew J. Colbrook, Athanasisos S. Fokas Recently a new transform method, called the Unified Transform or the Fokas method, for solving boundary value problems (BVPs) for linear and integrable nonlinear partial differential equations (PDEs) has received a lot of attention. For linear elliptic PDEs, this method yields two equations, known as the global relations, coupling the Dirichlet and Neumann boundary values. These equations can be used in a collocation method to determine the Dirichlet to Neumann map. This involves expanding the unknown functions in terms of a suitable basis, and choosing a set of collocation points at which to evaluate the global relations. Here, using these methods for the Helmholtz and modified Helmholtz equations and following the earlier results of [15], we determine eigenvalues of the Laplacian in a convex polygon. Eigenvalues are characterised by the points where the generalised Dirichlet to Neumann map becomes singular. We find that the method yields spectral convergence for eigenfunctions smooth on the boundary and for non-smooth boundary values, the rate of convergence is determined by the rate of convergence of expansions in the chosen Legendre basis. Extensions to the case of oblique derivative boundary conditions and constant coefficient elliptic PDEs are also discussed and demonstrated.

Authors:Marzieh Dehghani-Madiseh; Milan Hladík Pages: 18 - 33 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Marzieh Dehghani-Madiseh, Milan Hladík We investigate the interval generalized Sylvester matrix equation A X B + C X D = F . We propose a necessary condition for its solutions, and also a sufficient condition for boundedness of the whole solution set. The main effort is performed to develop techniques for computing outer estimations of the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator, reducing significantly computational complexity, compared to the Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of A, B, C and D and in both of them we suppose that the midpoints of A and C are simultaneously diagonalizable as well as for the midpoints of the matrices B and D. Numerical experiments are given to illustrate the performance of the proposed methods.

Authors:Jinwei Fang; Boying Wu; Wenjie Liu Pages: 34 - 52 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Jinwei Fang, Boying Wu, Wenjie Liu In this paper, we present a stable and efficient numerical scheme for the linearized Korteweg–de Vries equation on unbounded domain. After employing the Crank–Nicolson method for temporal discretization, the transparent boundary conditions are derived for the time semi-discrete scheme. Then the unconditional stability of the resulting initial boundary problem is established. For spatial discretization, we construct a non-polynomial based spectral collocation method in which the basis functions are built upon a generalized Birkhoff interpolation. The interpolation error of the new basis is also investigated. Moreover, the basis functions build in two free parameters intrinsically which can be chosen properly so that the implicit time semi-discrete scheme collapses to an explicit scheme after spatial discretization. Numerical tests are performed to demonstrate the stability and accuracy of the proposed method.

Authors:Mikael Barboteu; Krzysztof Bartosz; David Danan Pages: 53 - 77 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): Mikael Barboteu, Krzysztof Bartosz, David Danan We consider a dynamic process of frictional contact between a non-clamped viscoelastic body and a foundation. We assume that the normal contact response depends on the depth of penetration of the foundation by the considered body, and the dependence between these two quantities is governed by normal compliance conditions. On the other hand, the friction force is assumed to be a nonmonotone function of the slip rate where the friction threshold also depends on the depth of the penetration. Our aim in this paper is twofold. The first one is to prove the existence and the uniqueness of a weak solution for the contact problem under consideration. The second one is to provide the numerical analysis of the process involving its semi-discrete and fully discrete approximation as well as estimation of the error for both numerical schemes and the validation of such a result.

Authors:F. Dell'Accio; F. Di Tommaso; O. Nouisser; B. Zerroudi Pages: 78 - 91 Abstract: Publication date: April 2018 Source:Applied Numerical Mathematics, Volume 126 Author(s): F. Dell'Accio, F. Di Tommaso, O. Nouisser, B. Zerroudi In this paper we discuss an improvement of the triangular Shepard operator proposed by Little to extend the Shepard method. In particular, we use triangle based basis functions in combination with a modified version of the linear local interpolant on the vertices of the triangle. We deeply study the resulting operator, which uses functional and derivative data, has cubic approximation order and a good accuracy of approximation. Suggestions on how to avoid the use of derivative data, without losing both order and accuracy of approximation, are given.

Authors:Toby Sanders Pages: 1 - 9 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Toby Sanders Popular methods for finding regularized solutions to inverse problems include sparsity promoting ℓ 1 regularization techniques, one in particular which is the well known total variation (TV) regularization. More recently, several higher order (HO) methods similar to TV have been proposed, which we generally refer to as HOTV methods. In this letter, we investigate the problem of the often debated selection of λ, the parameter used to carefully balance the interplay between data fitting and regularization terms. We theoretically argue for a scaling of the operators for a uniform parameter selection for all orders of HOTV regularization. In particular, parameter selection for all orders of HOTV may be determined by scaling an initial parameter for TV, which the imaging community may be more familiar with. We also provide several numerical results which justify our theoretical findings.

Authors:Xiaojun Tang; Heyong Xu Pages: 51 - 67 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Xiaojun Tang, Heyong Xu The main purpose of this work is to develop an integral pseudospectral scheme for solving integro-differential equations. We provide new pseudospectral integration matrices (PIMs) for the Legendre–Gauss and the flipped Legendre–Gauss–Radau points, respectively, and present an efficient and stable approach to computing the PIMs via the recursive calculation of Legendre integration matrices. Furthermore, we provide a rigorous convergence analysis for the proposed pseudospectral scheme in both L ∞ and L 2 spaces via a linear integral equation, and the spectral rate of convergence is demonstrated by numerical results.

Authors:Weiyang Ding; Michael Ng; Yimin Wei Pages: 68 - 85 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Weiyang Ding, Michael Ng, Yimin Wei In this paper, we study a fast algorithm for finding stationary joint probability distributions of sparse Markov chains or multilinear PageRank vectors which arise from data mining applications. In these applications, the main computational problem is to calculate and store solutions of many unknowns in joint probability distributions of sparse Markov chains. Our idea is to approximate large-scale solutions of such sparse Markov chains by two components: the sparsity component and the rank-one component. Here the non-zero locations in the sparsity component refer to important associations in the joint probability distribution and the rank-one component refers to a background value of the solution. We propose to determine solutions by formulating and solving sparse and rank-one optimization problems via closed form solutions. The convergence of the truncated power method is established. Numerical examples of multilinear PageRank vector calculation and second-order web-linkage analysis are presented to show the efficiency of the proposed method. It is shown that both computation and storage are significantly reduced by comparing with the traditional power method.

Authors:Erge Ideon; Peeter Oja Pages: 143 - 158 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Erge Ideon, Peeter Oja We investigate the collocation method with quadratic/linear rational spline S of smoothness class C 2 for the numerical solution of two-point boundary value problems if the solution y (or −y) of the boundary value problem is a strictly convex function. We show that on the uniform mesh it holds ‖ S − y ‖ ∞ = O ( h 2 ) . Established bound of error gives a dependence on the solution of the boundary value problem and its coefficient functions. We prove also convergence rates ‖ S ′ − y ′ ‖ ∞ = O ( h 2 ) and ‖ S ″ − y ″ ‖ ∞ = O ( h 2 ) . Numerical examples support the obtained theoretical results.

Authors:Helmut Harbrecht; Michael D. Peters Pages: 159 - 171 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Helmut Harbrecht, Michael D. Peters The present article is dedicated to the solution of elliptic boundary value problems on random domains. We apply a high-precision second order shape Taylor expansion to quantify the impact of the random perturbation on the solution. Thus, we obtain a representation of the solution with third order accuracy in the size of the perturbation's amplitude. The major advantage of this approach is that we end up with purely deterministic equations for the solution's moments. In particular, we derive representations for the first four moments, i.e., expectation, variance, skewness and kurtosis. These moments are efficiently computable by means of boundary integral equations. Numerical results are presented to validate the presented approach.

Authors:Lin Mu; Junping Wang; Xiu Ye; Shangyou Zhang Pages: 172 - 182 Abstract: Publication date: March 2018 Source:Applied Numerical Mathematics, Volume 125 Author(s): Lin Mu, Junping Wang, Xiu Ye, Shangyou Zhang A discrete divergence free weak Galerkin finite element method is developed for the Stokes equations based on a weak Galerkin (WG) method introduced in [17]. Discrete divergence free bases are constructed explicitly for the lowest order weak Galerkin elements in two and three dimensional spaces. These basis functions can be derived on general meshes of arbitrary shape of polygons and polyhedrons. With the divergence free basis derived, the discrete divergence free WG scheme can eliminate pressure variable from the system and reduces a saddle point problem to a symmetric and positive definite system with many fewer unknowns. Numerical results are presented to demonstrate the robustness and accuracy of this discrete divergence free WG method.

Authors:Shimin Chai; Yanzhao Cao; Yongkui Zou; Wenju Zhao Pages: 44 - 56 Abstract: Publication date: February 2018 Source:Applied Numerical Mathematics, Volume 124 Author(s): Shimin Chai, Yanzhao Cao, Yongkui Zou, Wenju Zhao This paper is concerned with the finite element approximation of the stochastic Cahn–Hilliard–Cook equation driven by an infinite dimensional Wiener type noise. The Argyris finite elements are used to discretize the spatial variables while the infinite dimensional (cylindrical) Wiener process is approximated by truncated stochastic series spanned by the spectral basis of the covariance operator. The optimal strong convergence order in L 2 and H ˙ − 2 norms is obtained. Unlike the mixed finite element method studied in the existing literature, our method allows the covariance operator of the Wiener process to have an infinite trace, including the space–time white noise is allowed in our model. Numerical experiments are presented to illustrate the theoretical analysis.

Authors:Xiaoli Li; Hongxing Rui Pages: 89 - 109 Abstract: Publication date: February 2018 Source:Applied Numerical Mathematics, Volume 124 Author(s): Xiaoli Li, Hongxing Rui Based on the weighted and shifted Grünwald–Letnikov difference operator, a new high-order block-centered finite difference method is derived for the time-fractional advection–dispersion equation by introducing an auxiliary flux variable to guarantee full mass conservation. The stability and the global convergence of the scheme are proved rigorously. Some a priori estimates of discrete norms with optimal order of convergence O ( Δ t 3 + h 2 + k 2 ) both for solute concentration and the auxiliary flux variable are established on non-uniform rectangular grids, where Δ t , h and k are the step sizes in time, space in x- and y-direction. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Authors:Guanyu Zhou Pages: 1 - 21 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Guanyu Zhou We consider the fictitious domain method with H 1 -penalty for the Stokes problem with Dirichlet boundary condition. First, for the continuous penalty problem, we obtain the optimal error estimate O ( ϵ ) for both the velocity and pressure, where ϵ is the penalty parameter. Moreover, we investigate the H m -regularity for the solution of the penalty problem. Then, we apply the finite element method with the P1/P1 element to the penalty problem. Since the solution to the penalty problem has a jump in the traction vector, we introduce some interpolation/projection operators, as well as an inf-sup condition with the norm depending on ϵ. With the help of these preliminaries, we derive the error estimates for the finite element approximation. The theoretical results are verified by the numerical experiments.

Authors:R. Donat; F. Guerrero; P. Mulet Pages: 22 - 42 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): R. Donat, F. Guerrero, P. Mulet Chromatographic processes can be modeled by nonlinear, convection-dominated partial differential equations, together with nonlinear relations: the adsorption isotherms. In this paper we consider the nonlinear equilibrium dispersive (ED) model with adsorption isotherms of Langmuir type. We show that, in this case, the ED model can be written as a system of conservation laws when the dispersion coefficient vanishes. We also show that the function that relates the conserved variables and the physically observed concentrations of the components in the mixture is one to one and it admits a global inverse, which cannot be given explicitly, but can be adequately computed. As a result, fully conservative numerical schemes can be designed for the ED model in chromatography. We propose a Weighted-Essentially-non-Oscillatory second order IMEX scheme and describe the numerical issues involved in its application. Through a series of numerical experiments, we show that our scheme gives accurate numerical solutions which capture the sharp discontinuities present in the chromatographic fronts, with the same stability restrictions as in the purely hyperbolic case.

Authors:Stefano Cipolla; Fabio Durastante Pages: 43 - 57 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Stefano Cipolla, Fabio Durastante In this paper, using an optimize-then-discretize approach, we address the numerical solution of two Fraction Partial Differential Equation constrained optimization problems: the Fractional Advection Dispersion Equation (FADE) and the two-dimensional semilinear Riesz Space Fractional Diffusion equation. Both a theoretical and experimental analysis of the problem is carried out. The algorithmic framework is based on the L-BFGS method coupled with a Krylov subspace solver. A suitable preconditioning strategy by approximate inverses is taken into account. Graphics Processing Unit (GPU) accelerator is used in the construction of the preconditioners. The numerical experiments are performed with benchmarked software/libraries enforcing the reproducibility of the results.

Authors:Eduardo M. Garau; Rafael Vázquez Pages: 58 - 87 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Eduardo M. Garau, Rafael Vázquez In this article we introduce all the ingredients to develop adaptive isogeometric methods based on hierarchical B-splines. In particular, we give precise definitions of local refinement and coarsening that, unlike previously existing methods, can be understood as the inverse of each other. We also define simple and intuitive data structures for the implementation of hierarchical B-splines, and algorithms for refinement and coarsening that take advantage of local information. We complete the paper with some simple numerical tests to show the performance of the data structures and algorithms, that have been implemented in the open-source Octave/Matlab code GeoPDEs.

Authors:Jin Zhang; Xiaowei Liu Pages: 88 - 98 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Jin Zhang, Xiaowei Liu A continuous interior penalty method with piecewise polynomials of degree p ≥ 2 is applied on a Shishkin mesh to solve a singularly perturbed convection–diffusion problem, whose solution has a single boundary layer. This method is analyzed by means of a series of integral identities developed for the convection terms. Then we prove a supercloseness bound of order 5/2 for a vertex-cell interpolation when p = 2 . The sharpness of our analysis is supported by some numerical experiments. Moreover, numerical tests show supercloseness clearly for p ≥ 3 .

Authors:Farhad Fakhar-Izadi; Mehdi Dehghan Pages: 99 - 120 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Farhad Fakhar-Izadi, Mehdi Dehghan The numerical approximation of solution to nonlinear parabolic Volterra and Fredholm partial integro-differential equations is studied in this paper. Unlike the conventional methods which discretize the time variable by finite difference schemes, we use the spectral method for this purpose. Indeed, both of the space and time discretizations are based on the Legendre-collocation method which lead to conversion of the problem to a nonlinear system of algebraic equations. The convergence of the proposed method is proven by providing an L ∞ error estimate. Several numerical examples are included to demonstrate the efficiency and spectral accuracy of the proposed method in the space and time directions.

Authors:Pouria Assari; Mehdi Dehghan Pages: 137 - 158 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Pouria Assari, Mehdi Dehghan The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates.

Authors:S. Kopecz; A. Meister Pages: 159 - 179 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): S. Kopecz, A. Meister In [6] the modified Patankar–Euler and modified Patankar–Runge–Kutta schemes were introduced to solve positive and conservative systems of ordinary differential equations. These modifications of the forward Euler scheme and Heun's method guarantee positivity and conservation irrespective of the chosen time step size. In this paper we introduce a general definition of modified Patankar–Runge–Kutta schemes and derive necessary and sufficient conditions to obtain first and second order methods. We also introduce two novel families of two-stage second order modified Patankar–Runge–Kutta schemes.

Authors:Chenguang Zhou; Yongkui Zou; Shimin Chai; Qian Zhang; Hongze Zhu Pages: 180 - 199 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Chenguang Zhou, Yongkui Zou, Shimin Chai, Qian Zhang, Hongze Zhu In this paper, we apply a new weak Galerkin mixed finite element method (WGMFEM) with stabilization term to solve heat equations. This method allows the usage of totally discontinuous functions in the approximation space. The WGMFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. In addition, we develop and analyze the error estimates for both continuous and discontinuous time WGMFEM schemes. Optimal order error estimates in both L 2 and triple-bar ⫼ ⋅ ⫼ norms are established, respectively. Finally, numerical tests are conducted to illustrate the theoretical results.

Authors:Hanyu Li; Shaoxin Wang Pages: 200 - 220 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Hanyu Li, Shaoxin Wang The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be called the unified condition number, we first present an expression of the partial unified condition number when the data space is measured by a general weighted product norm. Then, by setting the specific norms and weight parameters, we obtain the expressions of the partial normwise, mixed and componentwise condition numbers. Moreover, the corresponding structured partial condition numbers are also taken into consideration when the problem is structured. Considering the connections between the indefinite and total least squares problems, we derive the (structured) partial condition numbers for the latter, which generalize the ones in the literature. To estimate these condition numbers effectively and reliably, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and three related algorithms are devised. Finally, the obtained results are illustrated by numerical experiments.

Authors:Arijit Hazra; Gert Lube; Hans-Georg Raumer Pages: 241 - 255 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Arijit Hazra, Gert Lube, Hans-Georg Raumer Magnetic Resonance Imaging (MRI) is a widely applied non-invasive imaging modality based on non-ionizing radiation which gives excellent images and soft tissue contrast of living tissues. We consider the modified Bloch problem as a model of MRI for flowing spins in an incompressible flow field. After establishing the well-posedness of the corresponding evolution problem, we analyze its spatial semi-discretization using discontinuous Galerkin methods. The high frequency time evolution requires a proper explicit and adaptive temporal discretization. The applicability of the approach is shown for basic examples.

Authors:María González; Virginia Selgas Pages: 275 - 299 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): María González, Virginia Selgas We consider a fluid–structure interaction problem consisting of the time-dependent Stokes equations in the fluid domain coupled with the equations of linear elastodynamics in the solid domain. For simplicity, all changes of geometry are neglected. We propose a new method in terms of the fluid velocity, the fluid pressure, the structural velocity and the Cauchy stress tensor. We show that the new weak formulation is well-posed. Then, we propose a new semidiscrete problem where the velocities and the fluid pressure are approximated using a stable pair for the Stokes problem in the fluid domain and compatible finite elements in the solid domain. We obtain a priori estimates for the solution of the semidiscrete problem, prove the convergence of these solutions to the solution of the weak formulation and obtain error estimates. A time discretization based on the backward Euler method leads to a fully discrete scheme in which the computation of the approximated Cauchy stress tensor can be decoupled from that of the remaining unknowns at each time step. The displacements in the structure (if needed) can be recovered by quadrature. Finally, some numerical experiments showing the performance of the method are provided.

Authors:Francesco Mezzadri; Emanuele Galligani Pages: 300 - 319 Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Francesco Mezzadri, Emanuele Galligani In this paper we solve a 2D nonlinear, non-steady reaction–convection–diffusion equation subject to Dirichlet boundary conditions by an iterative procedure consisting in lagging the diffusion term. First, we analyze the procedure, which we call Lagged Diffusivity Method. In particular, we provide a proof of the uniqueness of the solution and of the convergence of the lagged iteration when some assumptions are satisfied. We also describe outer and inner solvers, with special regard to how to link the stopping criteria in an efficient way. Numerical experiments are then introduced in order to evaluate the role of different linear solvers and of other components of the solution procedure, considering also the effect of the discretization.

Authors:Luigi Brugnano; Gianmarco Gurioli; Felice Iavernaro; Ewa B. Weinmüller Abstract: Publication date: Available online 24 December 2017 Source:Applied Numerical Mathematics Author(s): Luigi Brugnano, Gianmarco Gurioli, Felice Iavernaro, Ewa B. Weinmüller In this paper, we propose a second-order energy-conserving approximation procedure for Hamiltonian systems with holonomic constraints. The derivation of the procedure relies on the use of the so-called line integral framework. We provide numerical experiments to illustrate theoretical findings.

Authors:Jianguo Huang; Xuehai Huang Abstract: Publication date: Available online 21 December 2017 Source:Applied Numerical Mathematics Author(s): Jianguo Huang, Xuehai Huang An hp-version error analysis is developed for the general DG method in mixed formulation for solving the linear elastic problem. First of all, we give the hp-version error estimates of two L 2 projection operators. Then incorporated with the techniques in [11], we obtain the hp-version error estimates in energy norm and L 2 norm. Some numerical experiments are provided for demonstrating the theoretical results.

Authors:Olga V. Ushakova Abstract: Publication date: Available online 20 December 2017 Source:Applied Numerical Mathematics Author(s): Olga V. Ushakova The aim of the paper is to give the numerical criteria for classification of different types of hexahedral cells which can emerge in a three-dimensional structured grid generation. In general, computational grids and their cells have to be nondegenerate, however, in practice, situations arise in which degenerate grids are used and computed. In these cases, to prevent lost of accuracy, special strategies must be chosen both in grid generation and physical phenomenon solution algorithms. To determine which cells need a modification in above strategies, degenerate cells have to be detected. The criteria are suggested for hexahedral cells constructed by a trilinear mapping of the unit cube. All hexahedral cells are divided into nondegenerate and degenerate. Among nondegenerate hexahedral cells, cells exotic in shape are singled out as inadmissible. Degenerate cells are divided into pyramids, prisms and tetrahedrons—types of cells which can be admissible in grid generation and solution algorithms. Inadmissible types of degenerations are also considered. An algorithm for testing three-dimensional structured grids according to suggested criteria is described. Both results of testing and examples of different types of cells are demonstrated. In conclusion, recommendations for structured grid generation with the purpose to exclude undesirable types of cells are given.

Authors:Hua Wang; Jinru Chen; Pengtao Sun; Fangfang Qin Abstract: Publication date: Available online 20 December 2017 Source:Applied Numerical Mathematics Author(s): Hua Wang, Jinru Chen, Pengtao Sun, Fangfang Qin A new conforming enriched finite element method is presented for elliptic interface problems with interface-unfitted meshes. The conforming enriched finite element space is constructed based on the P 1 -conforming finite element space. Approximation capability of the conforming enriched finite element space is analyzed. The standard conforming Galerkin method is considered without any penalty stabilization term. Our method does not limit the diffusion coefficient of the elliptic interface problem to a piecewise constant. Finite element errors in H 1 -norm and L 2 -norm are proved to be optimal. Numerical experiments are carried out to validate theoretical results.

Authors:V.C. Le; H.T. Pham; T.T.M. Ta Abstract: Publication date: Available online 20 December 2017 Source:Applied Numerical Mathematics Author(s): V.C. Le, H.T. Pham, T.T.M. Ta In the context of structural optimization in fluid mechanics we propose a numerical method based on a combination of the classical shape derivative and Hadamard's boundary variation method. Our approach regards the viscous flows governed by Stokes equations with the objective function of energy dissipation and a constrained volume. The shape derivative is computed by Lagrange's approach via the solutions of Stokes and adjoint systems. The programs are written in FreeFem++ using the Finite Element method.

Authors:Rekha P. Kulkarni; Gobinda Rakshit Abstract: Publication date: Available online 18 December 2017 Source:Applied Numerical Mathematics Author(s): Rekha P. Kulkarni, Gobinda Rakshit Approximate solutions of linear and nonlinear integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we consider discrete versions of the modified projection method and of the iterated modified projection method for solution of a Urysohn integral equation with a smooth kernel. For r ≥ 1 , a space of piecewise polynomials of degree ≤ r − 1 with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at r Gauss points. The orders of convergence which we obtain for these discrete versions indicate the choice of numerical quadrature which preserves the orders of convergence. Numerical results are given for a specific example.

Authors:Xinpeng Yuan; Chunguang Xiong Guoqing Zhu Abstract: Publication date: Available online 14 December 2017 Source:Applied Numerical Mathematics Author(s): Xinpeng Yuan, Chunguang Xiong, Guoqing Zhu In this paper, employing ideas developed for conservation law equations such as the Lax–Friedrich-type and Godunov-type numerical fluxes, we describe the numerical schemes for approximating the solution of the limit problem arising in the homogenization of Hamilton–Jacobi equations. All approximation methods involve three steps. The first scheme is a provably monotonic discretization of the cell problem for approximating the effective Hamiltonian for a given vector P ∈ R N . Next, using interpolation, we present an approximation of the effective Hamiltonian in the domain R N . Finally, the numerical schemes of the Hamilton–Jacobi equations with the effective Hamiltonian approximation are constructed. We also present global error estimates including all the discrete mesh sizes. The theoretical results are illustrated through numerical examples, including two convex Hamiltonians and two non-convex Hamiltonians.

Authors:D.D. Tcheutia; A.S. Jooste; W. Koepf Abstract: Publication date: Available online 14 November 2017 Source:Applied Numerical Mathematics Author(s): D.D. Tcheutia, A.S. Jooste, W. Koepf Using the q-version of Zeilberger's algorithm, we provide a procedure to find mixed recurrence equations satisfied by classical q-orthogonal polynomials with shifted parameters. These equations are used to investigate interlacing properties of zeros of sequences of q-orthogonal polynomials. In the cases where zeros do not interlace, we give some numerical examples to illustrate this.

Authors:Tong Zhang; JiaoJiao Jin; YuGao HuangFu Abstract: Publication date: Available online 10 November 2017 Source:Applied Numerical Mathematics Author(s): Tong Zhang, JiaoJiao Jin, YuGao HuangFu In this paper, we consider the stability and convergence results of the Crank-Nicolson/Adams-Bashforth scheme for the Burgers equation with smooth and nonsmooth initial data. The spatial approximation is based on the standard conforming finite element space. The temporal treatment of the spatial discrete Burgers equation is based on the implicit Crank-Nicolson scheme for the linear term and the explicit Adams-Bashforth scheme for the nonlinear term. Firstly, we prove that the Crank-Nicolson/Adams-Bashforth scheme is almost unconditionally stable with initial data u 0 ∈ H α ( α = 1 , 2 ) . Secondly, the optimal error estimates of the numerical solution in L 2 -norm are derived with initial data u 0 ∈ H 2 , and the error estimates of approximate solution in L 2 norm obtained with initial data u 0 ∈ H 1 is reduced by 1 2 . Finally, some numerical examples are provided to verify the established stability theory and convergence results with H 2 and H 1 initial data.

Authors:Yongchao Yu; Jigen Peng Abstract: Publication date: Available online 3 November 2017 Source:Applied Numerical Mathematics Author(s): Yongchao Yu, Jigen Peng This paper studies algorithms for solving quadratically constrained ℓ 1 minimization and Dantzig selector which have recently been widely used to tackle sparse recovery problems in compressive sensing. The two optimization models can be reformulated via two indicator functions as special cases of a general convex composite model which minimizes the sum of two convex functions with one composed with a matrix operator. The general model can be transformed into a fixed-point problem for a nonlinear operator which is composed of a proximity operator and an expansive matrix operator, and then a new iterative scheme based on the expansive matrix splitting is proposed to find fixed-points of the nonlinear operator. We also give some mild conditions to guarantee that the iterative sequence generated by the scheme converges to a fixed-point of the nonlinear operator. Further, two specific proximal fixed-point algorithms based on the scheme are developed and then applied to quadratically constrained ℓ 1 minimization and Dantzig selector. Numerical results have demonstrated that the proposed algorithms are comparable to the state-of-the-art algorithms for recovering sparse signals with different sizes and dynamic ranges in terms of both accuracy and speed. In addition, we also extend the proposed algorithms to solve two harder constrained total-variation minimization problems.

Authors:Peter Minev; Petr N. Vabishchevich Abstract: Publication date: Available online 31 October 2017 Source:Applied Numerical Mathematics Author(s): Peter Minev, Petr N. Vabishchevich In this paper we consider various splitting schemes for unsteady problems containing the grad-div operator. The fully implicit discretization of such problems would yield at each time step a linear problem that couples all components of the solution vector. In this paper we discuss various possibilities to decouple the equations for the different components that result in unconditionally stable schemes. If the spatial discretization uses Cartesian grids, the resulting schemes are Locally One Dimensional (LOD). The stability analysis of these schemes is based on the general stability theory of additive operator-difference schemes developed by Samarskii and his collaborators. The results of the theoretical analysis are illustrated on a 2D numerical example with a smooth manufactured solution.

Authors:Michael Weise Abstract: Publication date: Available online 31 October 2017 Source:Applied Numerical Mathematics Author(s): Michael Weise Residual error estimation for conforming finite element discretisations of the isotropic Kirchhoff plate problem is covered by an estimator of Verfürth for the related biharmonic equation. This article generalises Verfürth's result to Kirchhoff plates with an anisotropic material, which requires some modifications. Special emphasis is laid on the reduced Hsieh–Clough–Tocher triangular finite element, the conforming element with the least possible number of unknowns.

Authors:G. Filipuk; M.N. Rebocho Abstract: Publication date: Available online 23 October 2017 Source:Applied Numerical Mathematics Author(s): G. Filipuk, M.N. Rebocho In this paper we study families of semi-classical orthogonal polynomials within class one. We derive general second or third order ordinary differential equations (with respect to certain parameters) for the recurrence coefficients of the three-term recurrence relation of these polynomials and show that in particular well-known cases, e.g. related to the modified Airy and Laguerre weights, these equations can be reduced to the second and the fourth Painlevé equations.

Authors:Donatella Occorsio; Maria Grazia Russo Abstract: Publication date: Available online 18 October 2017 Source:Applied Numerical Mathematics Author(s): Donatella Occorsio, Maria Grazia Russo Let w ( x ) = e − x β x α , w ¯ ( x ) = x w ( x ) and let { p m ( w ) } m , { p m ( w ¯ ) } m be the corresponding sequences of orthonormal polynomials. Since the zeros of p m + 1 ( w ) interlace those of p m ( w ¯ ) , it makes sense to construct an interpolation process essentially based on the zeros of Q 2 m + 1 : = p m + 1 ( w ) p m ( w ¯ ) , which is called “Extended Lagrange Interpolation". In this paper the convergence of this interpolation process is studied in suitable weighted L 1 spaces, in a general framework which completes the results given by the same authors in weighted L u p ( ( 0 , + ∞ ) ) , 1 ≤ p ≤ ∞ (see [30], [27]). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type ∫ 0 + ∞ f ( x ) k ( x , y ) u ( x ) d x , u ( x ) = e − x β x γ ( 1 + x ) λ , γ > − 1 , λ ∈ R + , where the kernel k ( x , y ) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [22]. It is shown that the “compound quadrature sequence” represents an efficient proposal for saving 1/3 of the evaluations of the function f, under unchanged speed of convergence. PubDate: 2017-10-25T12:06:02Z

Authors:Qili Tang; Yunqing Huang Abstract: Publication date: Available online 12 October 2017 Source:Applied Numerical Mathematics Author(s): Qili Tang, Yunqing Huang A fully discrete Crank-Nicolson leap-frog (CNLF) scheme is presented and studied for the nonstationary incompressible Navier-Stokes equations. The proposed scheme deals with the spatial discretization by Galerkin finite element method (FEM), treats the temporal discretization by CNLF method for the linear term and the semi-implicit method for nonlinear term. The almost unconditional stability, i.e., the time step is no more than a constant, is proven. By a new negative norm technique, the L 2 -optimal error estimates with respect to temporal and spacial orientation for the velocity are derived. At last, some numerical results are provided to justify our theoretical analysis.

Authors:Falletta Monegato; Scuderi Abstract: Publication date: February 2018 Source:Applied Numerical Mathematics, Volume 124 Author(s): S. Falletta, G. Monegato, L. Scuderi In this paper, we consider 3D wave propagation problems in unbounded domains, such as those of acoustic waves in non viscous fluids, or of seismic waves in (infinite) homogeneous isotropic materials, where the propagation velocity c is much higher than 1. For example, in the case of air and water c ≈ 343 m / s and c ≈ 1500 m / s respectively, while for seismic P-waves in linear solids we may have c ≈ 6000 m / s or higher. These waves can be generated by sources, possible away from the obstacles. We further assume that the dimensions of the obstacles are much smaller than that of the wave velocity, and that the problem transients are not excessively short. For their solution we consider two different approaches. The first directly uses a known space–time boundary integral equation to determine the problem solution. In the second one, after having defined an artificial boundary delimiting the region of computational interest, the above mentioned integral equation is interpreted as a non reflecting boundary condition to be coupled with a classical finite element method. For such problems, we show that in some cases the computational cost and storage, required by the above numerical approaches, can be significantly reduced by taking into account a property that till now has not been considered. To show the effectiveness of this reduction, the proposed approach is applied to several problems, including multiple scattering.

Authors:Xue Luo Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Xue Luo In this paper, we propose new spectral viscosity methods based on the generalized Hermite functions for the solution of nonlinear scalar conservation laws in the whole line. It is shown rigorously that these schemes converge to the unique entropy solution by using compensated compactness arguments, under some conditions. The numerical experiments of the inviscid Burger's equation support our result, and it verifies the reasonableness of the conditions.

Authors:Carlos Abstract: Publication date: January 2018 Source:Applied Numerical Mathematics, Volume 123 Author(s): Carlos Pérez-Arancibia This paper presents expressions for the classical combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems on the plane, in terms of smooth (continuously differentiable) integrands. These expressions are obtained by means of a singularity subtraction technique based on pointwise plane-wave expansions of the unknown density function. In particular, a novel regularization of the hypersingular operator is obtained, which, unlike regularizations based on Maue's integration-by-parts formula, does not give rise to involved Cauchy principal value integrals. Moreover, the expressions for the combined field integral operators and layer potentials presented in this contribution can be numerically evaluated at target points that are arbitrarily close to the boundary without severely compromising their accuracy. A variety of numerical examples in two spatial dimensions that consider three different Nyström discretizations for smooth domains and domains with corners—one of which is based on direct application of the trapezoidal rule—demonstrates the effectiveness of the proposed higher-order singularity subtraction approach.

Authors:Vít Dolejší; Georg May; Ajay Rangarajan Abstract: Publication date: Available online 2 October 2017 Source:Applied Numerical Mathematics Author(s): Vít Dolejší, Georg May, Ajay Rangarajan We present a continuous-mesh model for anisotropic hp-adaptation in the context of numerical methods using discontinuous piecewise polynomial approximation spaces. The present work is an extension of a previously proposed mesh-only (h-)adaptation method which uses both a continuous mesh, and a corresponding high-order continuous interpolation operator. In this previous formulation local anisotropy and global mesh density distribution may be determined by analytical optimization techniques, operating on the continuous mesh model. The addition of varying polynomial degree necessitates a departure from purely analytic optimization. However, we show in this article that a global optimization problem may still be formulated and solved by analytic optimization, adding only the necessity to solve numerically a single nonlinear algebraic equation per adaptation step to satisfy a constraint on the total number of degrees of freedom. The result is a tailorsuited continuous mesh with respect to a model for the global interpolation error measured in the L q -norm. From the continuous mesh a discrete triangular mesh may be generated using any metric-based mesh generator.

Authors:Simon Becher Abstract: Publication date: Available online 21 September 2017 Source:Applied Numerical Mathematics Author(s): Simon Becher We consider singularly perturbed boundary value problems with a simple interior turning point whose solutions exhibit an interior layer. These problems are discretised using higher order finite elements on layer-adapted piecewise equidistant meshes proposed by Sun and Stynes. We also study the streamline-diffusion finite element method (SDFEM) for such problems. For these methods error estimates uniform with respect to ε are proven in the energy norm and in the stronger SDFEM-norm, respectively. Numerical experiments confirm the theoretical findings.