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In this paper, we present the virtual element approximation for the modified transmission eigenvalue problem. The eigenvalue problem arises in inverse scattering theory for artificial inhomogeneous media characterized as a metamaterial. By the shifting argument and spectral approximation theory of compact operators, we prove the correct spectral approximation of the virtual element scheme, and derive the optimal order error estimates for the eigenfunctions and the double order for the eigenvalues. Finally, we implement numerical experiments to illustrate behaviors of the standard and serendipity virtual element schemes, and the finite element scheme. [ABSTRACT FROM AUTHOR]

The element free Galerkin technique is a meshless method based on the variational weak form in which the test and trial functions are the shape functions of moving least squares approximation. Since the shape functions of moving least squares approximation do not have the Kronecker property thus the Dirichlet boundary condition can not be applied directly and also in this case obtaining an error estimate is not simple. The main aim of the current paper is to propose an error estimate for the extracted numerical scheme from the element free Galerkin method. To this end, we select the fractional cable equation with Dirichlet boundary condition. Firstly, we obtain a time-discrete scheme based on a finite difference formula with convergence order O ( τ 1 + min ⁡ { α , β } ) , then we use the meshless element free Galerkin method, to discrete the space direction and obtain a full-discrete scheme. Also, for calculating the appeared integrals over the boundary and the domain of problem the Gauss–Legendre quadrature rule has been used. In the next, we change the main problem with Dirichlet boundary condition to a new problem with Robin boundary condition. Then, we show that the new technique is unconditionally stable and convergent using the energy method. We show convergence orders of the time discrete scheme and the full discrete scheme are O ( τ 1 + min ⁡ { α , β } ) and O ( r p + 1 + τ 1 + min ⁡ { α , β } ) , respectively. So, we can say that the main aim of this paper is as follows, (1) Transferring the main problem with Dirichlet boundary condition (old problem) to a problem with Robin boundary condition (new problem), (2) Showing that with special condition (when σ → + ∞ ) the solution of the new problem is convergent to the solution of the old problem, (3) Obtaining an error estimate for the new problem. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, the interpolating moving least-squares (IMLS) method is discussed in details. A simpler expression of the approximation function of the IMLS method is obtained. Compared with the moving least-squares (MLS) approximation, the shape function of the IMLS method satisfies the property of Kronecker δ function. Then the meshless method based on the IMLS method can overcome the difficulties of applying the essential boundary conditions. The error estimates of the approximation function and its first and second order derivatives of the IMLS method are presented in n -dimensional space. The theoretical results show that if the weight function is sufficiently smooth and the order of the polynomial basis functions is big enough, the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes. Then the interpolating element-free Galerkin (IEFG) method based on the IMLS method is presented for potential problems. The advantage of the IEFG method is that the essential boundary conditions can be applied directly and easily. For the purpose of demonstration, some selected numerical examples are given to prove the theories in this paper. [ABSTRACT FROM AUTHOR]

In this paper, a three-term conjugate gradient algorithm is developed for solving large-scale unconstrained optimization problems. The search direction at each iteration of the algorithm is determined by rectifying the steepest descent direction with the difference between the current iterative points and that between the gradients. It is proved that such a direction satisfies the approximate secant condition as well as the conjugacy condition. The strategies of acceleration and restart are incorporated into designing the algorithm to improve its numerical performance. Global convergence of the proposed algorithm is established under two mild assumptions. By implementing the algorithm to solve 75 benchmark test problems available in the literature, the obtained results indicate that the algorithm developed in this paper outperforms the existent similar state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Abstract In the paper, our main aim is to investigate the strong convergence of the implicit numerical approximations for neutral-type stochastic differential equations with super-linearly growing coefficients. After providing mean-square moment boundedness and mean-square exponential stability for the exact solution, we show that a backward Euler–Maruyama approximation converges strongly to the true solution under polynomial growth conditions for sufficiently small step size. Imposing a few additional conditions, we examine the p -th moment uniform boundedness of the exact and approximate solutions by the stopping time technique, and establish the convergence rate of one half, which is the same as the convergence rate of the classical Euler–Maruyama scheme. Finally, several numerical simulations illustrate our main results. [ABSTRACT FROM AUTHOR]

Abstract In this research, we give projection-based error analysis on a multiscale hybridizable discontinuous Galerkin method to numerically solve parabolic problem with a heterogeneous coefficient. We modified the spectral multiscale HDG method introduced in [22] to fit to the time-dependent PDE. The method uses multiscale spaces generated by eigenfunctions of local spectral problems. By considering two different grids, one relatively coarser than the other, we give bounds for the error between the actual solution and the approximate one derived from multiscale HDG method. One of the main focuses of the paper is to derive error analysis that depends on the size of fine and coarse grids and eigenvalues of local spectral problems. To solve a given coarse problem, the more eigenfunction we choose, the more accurate the approximation becomes: we shall see that the numerical result indicates that when we fix fine and coarse grids, the error between the reference solution and the derived one decreases whenever we have more multiscale basis functions. [ABSTRACT FROM AUTHOR]

Abstract Some standard numerical methods, such as mimetic finite difference method, finite volume method and mixed finite element method, often fail in solving nonlinear diffusion problems with degenerate diffusion coefficients, since a harmonic average of diffusion coefficients is involved in these methods. To avoid such problem, we present some finite volume element schemes to solve a class of 1D degenerate nonlinear parabolic equations in this paper. Some fully discrete schemes are given by using linear and quadratic finite volume elements in space and a backward difference formulation in time. To deal with unphysical numerical oscillation, we apply two nonnegativity-preserving repair techniques based on a posteriori corrections to finite volume element solutions. One is a local approach, in which any negative energy corresponding to some mesh node is absorbed by the positive values near the current node. The other is a global strategy, in which the total negative energy is redistributed to each positive value in the numerical solution in proportion to its value. In addition, some monotonous finite volume element schemes with lumped-mass strategy are presented. Numerical examples are included to demonstrate the effectiveness and competitive behavior of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton–Raphson–Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ , where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme. Highlights • A new transformation of Bratu's problem conserving its bifurcated behavior. • Computation of the two different branches of the solution for Bratu's problem. • An iterative numerical method based on Newton–Raphson–Kantorovich approximation. • A linear iterative scheme yielding accurate solutions of Bratu's problem. • Convergence of the method is proved to be quadratic, and its accuracy is confirmed numerically. [ABSTRACT FROM AUTHOR]

Abstract The paper provides a spectral collocation numerical scheme for the approximation of the solutions of stochastic fractional differential equations. The discretization of the operator leads to a system of nonlinear algebraic equations, whose coefficient matrix can be computed by an automatic procedure, consisting of linear steps. A selection of numerical experiments confirming the effectiveness of the approach is given, with respect to various sets of function bases and of collocation points. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract Ensemble methods, such as the ensemble Kalman filter (EnKF), the local ensemble transform Kalman filter (LETKF), and the ensemble Kalman smoother (EnKS) are widely used in sequential data assimilation, where state vectors are of huge dimension. Little is known, however, about the asymptotic behavior of ensemble methods. In this paper, we prove convergence in L p of ensemble Kalman smoother to the Kalman smoother in the large-ensemble limit, as well as the convergence of EnKS-4DVAR, which is a Levenberg–Marquardt-like algorithm with EnKS as the linear solver, to the classical Levenberg–Marquardt algorithm in which the linearized problem is solved exactly. [ABSTRACT FROM AUTHOR]

We describe a parallel implementation for the numerical approximation of solutions to the three-dimensional viscous, resistive magnetohydrodynamics (MHD) equations using a velocity–current formulation. In comparison to other formulations, the velocity–current formulation presented in this paper is an integro-differential system of equations that incorporates nonideal boundaries and nonlinearities due to induction. The solution to the equations is approximated using a Picard iteration, discretized with the finite element method, and solved iteratively with the Krylov subspace method GMRES. Effective preconditioning strategies are required to numerically solve the resulting equations with Krylov solvers [12] . For GMRES convergence, the system matrix resulting from the discretization of the velocity–current formulation is preconditioned using a simple, block-diagonal Schur-complement preconditioner based on [14] . The MHD solver is implemented using freely available, well-documented, open-source, libraries deal.II, p4est, Trilinos, and PETSc, capable of scaling to tens of thousands of processors on state-of-the-art HPC architectures. [ABSTRACT FROM AUTHOR]

In this paper, we develop a mixed Hermite–Fourier spectral method for the Fokker–Planck equation of periodic field. Radical to the numerical solutions of partial differential equations of the sort, results on mixed generalized Hermite–Fourier orthogonal approximation are established. The convergence of the constructed spectral scheme is proved. Numerical results show the efficiency of this approach and coincide well with theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form − div ( A ∇ u ) + V u + f ( u 2 ) u = λ u , ‖ u ‖ L 2 = 1 . We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f . In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this investigation, a computational scheme is given to solve nonlinear one- and two-dimensional Volterra integral equations of the second kind. We utilize the radial basis functions (RBFs) constructed on scattered points by combining the discrete collocation method to estimate the solution of Volterra integral equations. All integrals appeared in the scheme are approximately computed by the composite Gauss–Legendre integration formula. The implication of previous methods for solving these types of integral equations encounters difficulties by increasing the dimensional of problems and sometimes requires a mesh generation over the solution region. While the new technique presented in the current paper does not increase the difficulties for higher dimensional integral equations due to the easy adaption of RBF and also needs no cell structures on the domains. Moreover, we obtain the error bound and the convergence rate of the proposed approach. Illustrative examples clearly show the reliability and efficiency of the method and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical solution of the time fractional cable equation and its inverse problem. The time–space spectral Legendre tau method based on the shifted Legendre polynomial and its operational matrices is used to solve the direct problem. Furthermore, we prove that the approximated solution of this method converges to the exact solution. In addition, the inverse problem is formulated by using the Tikhonov regularization, the stability and convergence for the inverse problem are provided, then we analyse the sensitivity coefficients and apply the nonlinear conjugate gradient method to solve the regularized problem. Finally, some numerical results are carried out to support the theoretical claims. [ABSTRACT FROM AUTHOR]

In this paper, we study a spectral method for the triangular prism. We construct an approximation space in the “pole” condition in which the integral singularity is removed in a simple and effective way. We build a quasi-interpolation operator in the approximation space, and analyze its L 2 -error. Based on the quasi-interpolation, a triangular prism spectral method for the elliptic modal problem is studied. Furthermore, we extend this triangular prism spectral method to a triangular prism spectral element method. For the elliptic modal problem, we present the spectral element scheme and analyze the convergence. At last, we do some experiments to test the effectiveness of the method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 [31] , [28] ). 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 [23] . 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. [ABSTRACT FROM AUTHOR]

In this paper, we explore an efficient preconditioning method for the saddle point system resulting from a four-field mixed finite element method applied to Biot's consolidation model. The proposed preconditioner is a block diagonal preconditioner based on the Schur complement. We obtain bounds on the eigenvalues of the preconditioned matrix that are clustered away from 0. To reduce the computational expense, this preconditioner is inverted approximately. Some numerical results are provided to show the efficiency of our preconditioning strategy when applied to a poroelasticity problem in a layered medium. [ABSTRACT FROM AUTHOR]

In this paper, we present a space–time Legendre–Gauss–Lobatto (LGL) collocation method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions. The proposed method is based on the LGL collocation method to discretize in space, then use the LGL collocation method or block LGL collocation method to discretize in time. Our formulation has high-order accuracy in both space and time. We introduce a new H 1 projection for the error analysis of collocation method for nonhomogeneous Dirichlet boundary conditions. This projection differs from the classical H 1 projection. The new H 1 projection is identically equal to Lagrange interpolation on boundary. The approximation properties of the new H 1 projection are obtain. We derive error bounds in both discrete L 2 and H 1 norms for the (spatially) semidiscrete formulation, the analysis is based on new H 1 projection results. [ABSTRACT FROM AUTHOR]

In this paper, the fractional cable equation, involving two Riemann–Liouville fractional derivatives, with initial/boundary condition is considered. Two fully discrete schemes are obtained by employing piecewise linear Galerkin FEM in space, and using convolution quadrature methods based on the first- and second-order backward difference methods in time. Optimal error estimates in terms of the initial data and the inhomogeneity for the semi-discrete scheme and fully discrete schemes are discussed. Numerical results are shown to verify the theoretical results. [ABSTRACT FROM AUTHOR]

An efficient two-level finite element algorithm for solving the natural convection equations is developed and studied in this paper. By solving one small nonlinear system on a coarse mesh H and two large linearized problems on a fine mesh h = O ( H 7 − ε 2 ) with different loads, we can obtain an approximation solution ( u h , p h , T h ) with the convergence rate of same order as the usual finite element solution, which involves one large nonlinear natural convection system on the same fine mesh h . Furthermore, compared with the results of Si's algorithm in 2011, the given algorithm costs less computed time to get almost the same precision. [ABSTRACT FROM AUTHOR]

Motivated by recent stability results on one-step methods, especially Runge–Kutta methods, for the generalized pantograph equation (GPE), in this paper we study the stability of one-leg multistep methods for these equations since the one-leg methods have less computational cost than Runge–Kutta methods. To do this, a new stability concept, G q ( q ¯ ) -stability defined for variable stepsizes one-leg methods with the stepsize ratio q which is an extension of G -stability defined for constant stepsizes one-leg methods, is introduced. The Lyapunov functional of linear system is obtained and numerically approximated. It is proved that a G q ( q ¯ ) -stable fully-geometric mesh one-leg method can preserve the decay property of the Lyapunov functional for any q ∈ [ 1 , q ¯ ] . The asymptotic contractivity, a new stability concept at vanishing initial interval, is introduced for investigating the effect of the initial interval approximation on the stability of numerical solutions. This property and the bounded stability of G q ( q ¯ ) -stable one-leg methods for linear and nonlinear problems are analyzed. A numerical example which further illustrates our theoretical results is provided. [ABSTRACT FROM AUTHOR]

In this paper, efficient and computationally attractive methods based on the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations for the numerical solution of a system of Volterra integral equations with weakly singular kernels are presented. Simplicity for performing even in the presence of singularities is one of the advantages of Sinc methods. Convergence analysis of the proposed methods is given and an exponential convergence is achieved as well. Numerical results are presented which demonstrate the efficiency and high accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

The standard moving least squares (MLS) method might have an expensive computational cost when the number of test points and the dimension of the approximation space are large. To reduce the computational cost, this paper proposes a piece-wise moving least squares approximation method (PMLS) for scattered data approximation. We further apply the PMLS method to solve time-dependent partial differential equations (PDE) numerically. It is proven that the PMLS method is an optimal design with certain localized information. Numerical experiments are presented to demonstrate the efficiency and accuracy of the PMLS method in comparison with the standard MLS method in terms of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O ( Δ t 2 − α ) , 0 < α < 1 , where α is the order of the fractional derivative and Δ t is the step size. We then use a similar idea to prove the error estimates of the high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37] , where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O ( Δ t 3 − α ) , 0 < α < 1 . Numerical examples are given to show that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform of a given function f H p ( f w α , t ) = d p d t p ⨍ 0 + ∞ f ( x ) x − t w α ( x ) d x = p ! ⨎ 0 + ∞ f ( x ) ( x − t ) p + 1 w α ( x ) d x , where p ∈ { 1 , 2 , … } , t > 0 , and w α ( x ) = e − x x α is a Laguerre weight. The right-hand integral is defined as the finite part in the Hadamard sense. The proposed numerical approach is convenient when the approximation of the function H p ( f w α , t ) is required. Moreover, if there is the need, all the computations can be performed without differentiating the density function f . Numerical stability and convergence are proved in suitable weighted uniform spaces and numerical tests which confirm the theoretical estimates are presented. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is proposed to approximate the solution of a two-dimensional scattering problem of time-harmonic elastic wave from a rigid obstacle. By Helmholtz decomposition, the scattering problem is reduced to a system of Helmholtz equations with coupled boundary conditions. Then, we prove that the system of Helmholtz equations has only one solution under certain conditions, and propose an integral equation method to solve it numerically based on Tikhonov regularization method. Finally, numerical examples are presented to show the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The problem of consistency of smoothed particle hydrodynamics (SPH) has demanded considerable attention in the past few years due to the ever increasing number of applications of the method in many areas of science and engineering. A loss of consistency leads to an inevitable loss of approximation accuracy. In this paper, we revisit the issue of SPH kernel and particle consistency and demonstrate that SPH has a limiting second-order convergence rate. Numerical experiments with suitably chosen test functions validate this conclusion. In particular, we find that when using the root mean square error as a model evaluation statistics, well-known corrective SPH schemes, which were thought to converge to second, or even higher order, are actually first-order accurate, or at best close to second order. We also find that observing the joint limit when N → ∞ , h → 0 , and n → ∞ , as was recently proposed by Zhu et al., where N is the total number of particles, h is the smoothing length, and n is the number of neighbor particles, standard SPH restores full C 0 particle consistency for both the estimates of the function and its derivatives and becomes insensitive to particle disorder. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of severely ill-posed backward problems for linear parabolic equations. We use a convolution regularization method to obtain a stable approximate initial data from the noisy final data. The convergence rates are obtained under an a priori and an a posteriori regularization parameter choice rule in which the a posteriori parameter choice is a new generalized discrepancy principle based on a modified version of Morozov's discrepancy principle. The log-type convergence order under the a priori regularization parameter choice rule and log ⁡ log -type order under the a posteriori regularization parameter choice rule are obtained. Two numerical examples are tested to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient spectral method based on the Legendre–Galerkin approximation for the transmission eigenvalue problem. A rigorous error analysis is presented by using the minmax principle for the generalized eigenvalue problems associated to a transmission eigenvalue problem. However, this formulation can only compute real eigenvalues. Thus, we also present another formulation based on second order equations and construct an appropriate set of basis functions such that the matrices in the discrete variational form are sparse. For the case of constant medium, we derive the matrix formulations based on the tensor-product for the discrete variational form in two and three-dimensional cases, respectively. In addition, we also establish an optimization scheme based on the Legendre–Galerkin approximation. With this scheme we can estimate the index of refraction of an inhomogeneous medium. We also present ample numerical results to show that our method is very effective and high accurate. [ABSTRACT FROM AUTHOR]

In the present paper we introduce new block extrapolation methods as generalizations of the well known vector extrapolation methods. We give expressions of the obtained approximations via the Schur complement and also propose an efficient implementation of these methods. Applications to linearly generated sequences are given and extensions to nonlinear problems are also given. Applications of the proposed block extrapolation methods to some nonlinear matrix equations are considered and some numerical examples are given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive and evaluate new approximations of the Dirichlet-to-Neumann (DtN) and Neumann-to-Dirichlet (NtD) maps for two-dimensional acoustic scattering problems. Some formal approximations for the two-dimensional case are derived. These various approximations are next numerically validated and compared. [ABSTRACT FROM AUTHOR]

In this paper, a high-order compact alternating direction implicit (HOC ADI) method, which combines fourth-order compact difference approximation to spatial derivatives and second order backward differentiation formula (BDF2) for temporal integration, is derived for nonlinear delay reaction–diffusion equations. By the discrete energy method, its optimal error estimates in L 2 - and H 1 -norms are constructively obtained. Then, a class of Richardson extrapolation algorithms (REAs) are established to improve computational efficiency. Besides, a modified HOC ADI solver is devised to reduce time cost as delay is very small. Numerical results confirm the theoretical results and performance of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the mathematical analysis of a method based on fictitious domain approach. Boundary conditions of Robin type (also known as Fourier boundary conditions) are enforced using a penalization method. A complete description of the method and a full analysis are provided for univariate elliptic and parabolic problems using finite difference approximation. Numerical evidence of the predicted estimations is provided as well as numerical results for a nonlinear problem and a first extension of the method in the bivariate situation is proposed. [ABSTRACT FROM AUTHOR]

In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrange multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations. [ABSTRACT FROM AUTHOR]

In this paper, we develop a direct spectral method based on the mixed Laguerre–Legendre quasi-orthogonal approximation for non-isotropic heat transfer with inhomogeneous Neumann boundary condition in an infinite strip. This method guarantees that the homogeneous boundary condition is exactly satisfied, which differs from other spectral methods for Neumann problems. For analyzing the numerical errors, some basic results on the mixed Laguerre–Legendre quasi-orthogonal approximation are established. The convergence of the proposed scheme is proved. Numerical results demonstrate the efficiency of this new approach and coincide well with the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a class of hybrid difference schemes with variable weights on Bakhvalov–Shishkin mesh is proposed to compute both the solution and the derivative in quasilinear singularly perturbed convection–diffusion boundary value problems. The parameter-uniform second-order convergence of approximating to the solution and the derivative on Bakhvalov–Shishkin mesh and that of nearly second-order on Shishkin mesh are proved clearly by use of an ( l ∞ , l 1 ) -stability property, where the former sufficient conditions for uniform convergence are modestly relaxed on Bakhvalov–Shishkin mesh and are clarified on Shishkin mesh. The numerical examples support the proposed schemes with new sufficient conditions and their error estimates. [ABSTRACT FROM AUTHOR]

In this paper, a new nonmonotone adaptive trust region method with line search for solving unconstrained nonlinear optimization problems is introduced. The computation of the Hessian approximation is based on the usage of the weak secant equation by a diagonal definite matrix. Under some reasonable conditions, the global convergence of the proposed algorithm is established. The numerical results show the new method is effective and attractive for large scale optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we study the superconvergence properties of the discontinuous Galerkin (DG) method applied to one-dimensional m th-order ordinary differential equations without introducing auxiliary variables. We show that the leading term of the discretization error on each element is proportional to a combination of Jacobi polynomials. Thus, the p -degree DG solution is O ( h p + 2 ) superconvergent at the roots of specific combined Jacobi polynomials. Moreover, we use these results to compute simple, efficient and asymptotically exact a posteriori error estimates and to construct higher-order DG approximations. [ABSTRACT FROM AUTHOR]

We consider the discontinuous Qk-finite element approximations to the elliptic boundary value problems in d-dimensional rectangular domain. A derivative recovery technique is proposed by interpolating the derivatives of discrete solution on the patch domain. Based on the superclose estimate derived in this paper, we show that the recovered derivatives possess the local and global superconvergence. Furthermore, the asymptotically exact a posteriori estimator is given on the error of gradient approximation. Finally, numerical experiments are presented to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper presents a multiscale analysis for stochastic elliptic equations in heterogeneous media. The main contributions are threefold: derive the convergence rate of the first-order asymptotic solution based on the periodic approximation method; develop a new technique for dealing with a large stochastic fluctuation; and present a novel multiscale asymptotic method. A multiscale finite element method is developed, and numerical results for solving stochastic elliptic equations with rapidly oscillating coefficients are reported. [ABSTRACT FROM AUTHOR]