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We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties. We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class. The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish approximate representations of solutions to stochastic Volterra integral equations (SVIEs, for short), by virtue of solutions to a family of stochastic differential equations. As an application, we present two algorithms for solving SVIEs: stochastic theta method and splitting method, and prove the global 1/2 order convergence rates in L 2 norm. [ABSTRACT FROM AUTHOR]

In this paper, we present spectral methods in order to solve wave equation subject to a locally distributed nonlinear damping. Thanks to the efficiency and the accuracy of spectral method, we can check that discrete energy decreases to zero as time goes to infinity, uniformly with respect to the mesh size when the damping is supported in a suitable subset of the domain of consideration. We prove the convergence of the full Fourier–Galerkin discretization. Thus, we apply our schemes to illustrate the uniform discrete energy decay rates of the solution for a wide range of damping functions. [ABSTRACT FROM AUTHOR]

We prove that if the numerical range of a Hilbert space contraction T is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then ‖ T n ( I − T ) ‖ = O ( 1 / n β ) with β ∈ [ 1 2 , 1 ) . For normal contractions the condition is also necessary. Another sufficient condition for β = 1 2 , necessary for T normal, is that the numerical range of T be in a disk { z : | z − δ | ≤ 1 − δ } for some δ ∈ ( 0 , 1 ) . As a consequence of results of Seifert, we obtain that a power-bounded T on a Hilbert space satisfies ‖ T n ( I − T ) ‖ = O ( 1 / n β ) with β ∈ ( 0 , 1 ] if and only if sup 1 < | λ | < 2 ⁡ | λ − 1 | 1 / β ‖ R ( λ , T ) ‖ < ∞ . When T is a contraction on L 2 satisfying the numerical range condition, it is shown that T n f / n 1 − β converges to 0 a.e. with a maximal inequality, for every f ∈ L 2 . An example shows that in general a positive contraction T on L 2 may have an f ≥ 0 with lim sup T n f / log ⁡ n n = ∞ a.e. [ABSTRACT FROM AUTHOR]

In this paper, we prove the convergence of the numerical approximations of a scalar parabolic equation modeling a non-Newtonian fluid. We use finite-difference schemes and the well-known method of external approximations. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the second-order backward Euler scheme for a class of nonlinear expitaxial growth model. The difference scheme is three-level and can achieve second-order convergency in time and space. The unique solvability, unconditional stability and convergency in discrete-norm are strictly proved. Numerical examples are also given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

We present an asynchronous method for the explicit integration of multi-scale partial differential equations. This method is restricted by a local CFL condition rather than the traditional global CFL condition. First, we developed an upwind asynchronous forward Euler scheme for the transport equation and we proved that the asynchronous scheme is first order convergent. To improve the convergence rate of the asynchronous scheme, we derived an asynchronous Runge–Kutta 2 scheme from a standard explicit Runge–Kutta method. [ABSTRACT FROM AUTHOR]

In this paper, a modified Liu–Storey conjugate gradient method is proposed. The method can generate sufficient descent directions for non-linear unconstrained optimization problems. A global convergence result is established when the line search fulfils the strong Wolfe conditions. Moreover, the-linear convergence rate of the methods is proved. The extensive numerical results show that the proposed method is efficient for the unconstrained optimization problems in the CUTEr library. [ABSTRACT FROM AUTHOR]

Based on replacing integrals with quadrature and cubature formulas, we propose an algorithm for numerical solution of the linear Volterra equation with partial integrals and prove the convergence of this algorithm. [ABSTRACT FROM AUTHOR]

A new parametric class of third-order iterative methods for solving nonlinear equations and systems is presented. These schemes are showed to be more stable than Newton’, Traub’ or Ostrowski’s procedures (in some specific cases), and it has been proved that the set of starting points that converge to the roots of different nonlinear functions is wider than the one of those respective methods. Moreover, the numerical efficiency has been checked through different numerical tests. [ABSTRACT FROM AUTHOR]

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function. [ABSTRACT FROM AUTHOR]

Recently, some authors (Darvishi and Hessari, 2011; Zhang et al., 2012) discussed the generalized modified symmetric successive over relaxation (GMSSOR) method for saddle point problems. In this paper, we further study the GMSSOR method for solving singular saddle point problems. We prove the semi-convergence of the GMSSOR method without the assumption that the (1, 1)-block sub-matrix A should be symmetric, and analyze the spectral property of the corresponding preconditioned matrix. Numerical experiments are given to illustrate the efficiency of GMSSOR with appropriate parameters. [ABSTRACT FROM AUTHOR]

Abstract: A mass conservative multirate time integration method is developed for a time-dependent isothermal single-phase free flow system coupled with an isothermal single-fluid-phase porous medium system. Long time stability of the proposed numerical scheme is proved under a time step restriction which depends on the physical parameters of the flow systems and the ratio between the time steps applied in the free flow and porous medium domains. Convergence analysis is carried out and a priori error estimates are obtained. Numerical results are presented for two model problems and a realistic setup. Advantages of using multiple-time-step techniques for modeling coupled systems where the processes evolve on different time scales are demonstrated and discussed. [Copyright &y& Elsevier]

An approximately globally convergent numerical method proposed by Beilina and Klibanov for a coefficient inverse problem related to the hyperbolic equationis studied. While the global convergence of this method has been proved for thecase, incase, it was proved only partially. The last case is of an interest, since it was demonstrated that theversion of this method works well for a set of experimental data. In this paper, a complete proof of convergence of this method inis presented. [ABSTRACT FROM PUBLISHER]

We present error analysis of fully discrete Galerkin finite element methods for the time-dependent Ginzburg–Landau equations with the temporal gauge, where a linearized backward Euler scheme is used for the time discretization. We prove that the convergence rate isO(τ+hr) if the finite element space of piecewise polynomials of degreeris used. Due to the degeneracy of the problem, the convergence rate is one order lower than the optimal convergence rate of finite element methods for parabolic equations. Numerical examples are provided to support our theoretical analysis. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, for an -strict pseudocontraction T, we prove strong convergence of the modified Mann’s iteration defined by where and in satisfy: [(i)] with ; [(ii)] and ; [(iii)] . Our results unify and improve some existing results. [Copyright &y& Elsevier]

Abstract: The aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space. General convergence theorems are proven. Some particular cases of the general algorithm are introduced and demonstrated. Theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests. [Copyright &y& Elsevier]

In this paper, three preconditioned parallel multisplitting methods are established for solving the large sparse linear systems, including local preconditioned parallel multisplitting relaxation method, global preconditioned parallel multisplitting relaxation method and global preconditioned nonstationary parallel multisplitting relaxation method. The convergence and comparison results of the methods associated with USAOR multisplitting are given when the coefficient matrices of the linear systems are H-matrices. We prove three preconditioned parallel multisplitting USAOR method are better than parallel multisplitting USAOR methods for M-matrices linear systems. Finally, numerical examples are given to illustrate the methods are valid. [ABSTRACT FROM AUTHOR]

Abstract: We study the superconvergence property of the local discontinuous Galerkin (LDG) method for solving the linearized Korteweg–de Vries (KdV) equation. We prove that, if the piecewise polynomials with are used, the LDG solution converges to a particular projection of the exact solution with the order , when the upwind flux is used for the convection term and the alternating flux is used for the dispersive term. Numerical examples are provided at the end to support the theoretical results. [Copyright &y& Elsevier]

Abstract: This paper deals with the null controllability of an initial-boundary value problem for a parabolic coupled system with nonlinear terms of local and nonlocal kinds. The control is distributed in space and time and is exerted through one scalar function whose support can be arbitrarily small. We first prove that, if the initial data are sufficiently small and the linearized system at zero satisfies an appropriate coupling condition, the equations can be driven exactly to zero. We also present an iterative algorithm of the quasi-Newton kind for the computation of the control and we prove a convergence result. The behavior of this algorithm is illustrated with some numerical experiments. [Copyright &y& Elsevier]

In this article, we study the complete convergence for weighted sums of extended negatively dependent random variables and row sums of arrays of rowwise extended negatively dependent random variables. We apply two methods to prove the results: the first of is based on exponential bounds and second is based on the generalization of the classical moment inequality for extended negatively dependent random variables. [ABSTRACT FROM PUBLISHER]

The attraction of dual trajectories of Newton's method for the Lagrange system to critical Lagrange multipliers is analyzed. This stable effect, which has been confirmed by numerical practice, leads to the Newton-Lagrange method losing its superlinear convergence when applied to problems with irregular constraints. At the same time, available theoretical results are of 'negative' character; i.e., they show that convergence to a noncritical multiplier is not possible or unlikely. In the case of a purely quadratic problem with a single constraint, a 'positive' result is proved for the first time demonstrating that the critical multipliers are attractors for the dual trajectories. Additionally, the influence exerted by the attraction to critical multipliers on the convergence rate of direct and dual trajectories is characterized. [ABSTRACT FROM AUTHOR]

In this paper, a parametric accelerated Steffensen's method and three doubly-accelerated Steffensen's methods with memory are designed for solving nonlinear equations efficiently. Their orders of convergence from 2.414 up to 2.831 and 3 are proved theoretically and demonstrated numerically. Each of the accelerated Steffensen's methods only uses two new evaluations of the function without derivatives per step and is applicable on solving systems of nonlinear equations and nonlinear ODEs. [ABSTRACT FROM AUTHOR]

In this article we introduce the sequence space c0I(F, p, ∆nν) and ℓ∞I(F, p, ∆nν) for the of sequence of modulii F = (fk) and given some inclusion relations. These results here proved are analogus to those by M.Aiyub [1](Global Journal of Science Frontier Research Mathematics and Decision Sciences 12(9)(2012),32-36). [ABSTRACT FROM AUTHOR]

In this work, we present two modifications of an earlier presented iterative method for solving nonlinear equations in the case of multiple zeros. The new methods are two-point, derivative free iterative methods for the simultaneous extraction of one or several of the multiple zeros. We prove that the presented methods have quadratic convergence locally. Some numerical results with various nonlinear equations are given. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a low-order stabilized discretization of the primitive equations of the ocean with highly reduced computational complexity. We prove stability through a specific inf–sup condition, and weak convergence to a weak solution. We also perform some numerical tests for relevant flows. [Copyright &y& Elsevier]

Abstract: In this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations (FODEs). This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In our approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps, while preserving the good stability property of the block-by-block schemes. By using this new approach, we are able to construct a high order schema for FODEs of the order . The stability and convergence of the schema is rigorously established. We prove that the numerical solution converges to the exact solution with order for , and order 4 for α>1. A series of numerical examples are provided to support the theoretical claims. [Copyright &y& Elsevier]

Abstract: In this paper, we present a fast and efficient numerical method to solve a two-dimensional fractional evolution equation on a finite domain. This numerical method combines the alternating direction implicit (ADI) approach with the second-order difference quotient in space, the backward Euler in time and order one convolution quadrature approximating the integral term. By using the discrete energy method, we prove that the ADI scheme is unconditionally stable and the numerical solution converges to the exact one with order , where k is the temporal grid size and are spatial grid sizes in the x and y directions, respectively. Two numerical examples with known exact solution are also presented, and the behavior of the error is analyzed to verify the order of convergence of the ADI-Euler method. [Copyright &y& Elsevier]

Combining the numerical concept of variational discretization and semi-smooth Newton methods for the numerical solution of pde-constrained optimization with control constraints, we place special emphasis on the implementation and globalization of the numerical algorithm. We prove fast local convergence of a globalized algorithm and illustrate our analytical and algorithmical findings by numerical experiments. [ABSTRACT FROM AUTHOR]

The spectral methods of solving boundary value problems for mixed-type differential equations of second order in a 3D domain are studied. Existence and uniqueness of a solution of the Lavrent'ev–Bitsadze problem is proved. [ABSTRACT FROM AUTHOR]

This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initial-and boundary-value problems associated with the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in L∞ and the advection vector field v has low regularity properties, namely v ∈ L¹ (]0, T[, (W¹, ¹ (Q))d), with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in LP(Q), with p < +∞, of the approximate solution towards the unique weak solu-tion of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory. Note that this result remains valid if the data are suitably approximated in L¹. This is nothing but the discrete coun-terpart of the nice compactness properties deduced from the renormalized solution theory. We conclude with some numerical experiments showing that the convergence rate seems to be 1/2, as in the case of smoother advection fields, but this is still an open question up on up [ABSTRACT FROM AUTHOR]

Abstract: We investigate the inside structure of one-dimensional reaction–diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established. [Copyright &y& Elsevier]

Abstract: An approximation problem of parametric curves and surfaces is studied by a new kind of spline functions from some Lagrange or Hermite data set. We present an interpolation problem by minimizing a functional on a parametric finite element space in order to obtain the new notion of a spline. We call it discrete interpolating variational spline. We show how to compute in practice such spline and we carefully prove a convergence result. To illustrate the generality and practice of this work we give some particular cases and we finish by presenting some numerical and graphical examples. [Copyright &y& Elsevier]

In this paper, solving fuzzy ordinary differential equations of the nth order by Runge-Kutta method hav been done. In this following, the convergence of the proposed method is proved. The proposed method is illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, by approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders 4 and 6, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method. [Copyright &y& Elsevier]

This paper considers the multidimensional active scalar problem of motion of a function ρ(x, t) by a velocity field obtained by v = -∇N * ρ, where N is the Newtonian potential. We prove well-posedness of compactly supported L∞ ∩ L1 solutions of possibly mixed sign. These solutions include an important class of solutions that are proportional to characteristic functions on a time-evolving domain. We call these aggregation patches. Whereas positive solutions collapse on themselves in finite time, negative solutions spread and converge toward a self-similar spreading circular patch solution as t → ∞. We give a convergence rate that we prove is sharp in 2D. In the case of positive collapsing solutions, we investigate numerically the geometry of patch solutions in 2D and in 3D (axisymmetric). We show that the time evolving domain on which the patch is supported typically collapses on a complex skeleton of codimension one. [ABSTRACT FROM AUTHOR]

Abstract: A new two-point iterative method for solving nonlinear equations is presented. This is a derivative-free method for the simultaneous determination of several or all of the simple zeros. It is proved that the proposed method possesses a quadratic convergence locally. Numerical examples are given to illustrate the efficiency and performance of the method presented. [Copyright &y& Elsevier]

Abstract: Recently, Zhang take a little modification to the Wei–Yao–Liu nonlinear conjugate gradient method proposed by Wei et al. such that the modified method (called NPRP method in Zhang ) satisfies sufficient descent condition with greater parameter in the strong Wolfe line search and converges globally for nonconvex minimization with the strong Wolfe line search. In this paper, we take a little modification to the NPRP method such that the modified NPRP method possesses better properties than the NPRP method in Zhang . Firstly, the modified NPRP method possesses the sufficient descent property without any line searches. Secondly, the modified NPRP method converges globally for nonconvex minimization with Wolfe line search or Armijo line search. Moreover, we extend these results to the Hestenes–Stiefel (HS) method and prove that the modified HS method also possesses sufficient descent property and global convergence with the standard Wolfe conditions. Numerical results are reported by utilizing some test problems in the CUTE library. [Copyright &y& Elsevier]

For the numerical solution of boundary-value problems on the semiaxis for second-order nonlinear ordinary differential equations, an exact three-point difference scheme is constructed and substantiated. Under the conditions of existence and uniqueness of solution of a boundary-value problem, we prove the existence and uniqueness of solution of the exact three-point difference scheme and convergence of the method of successive approximations for its solution. [ABSTRACT FROM AUTHOR]

A numerical method possessing the approximate global convergence property is developed for a 3-D coefficient inverse problem for hyperbolic partial differential equations with backscattering data resulting from a single measurement. An important part of this technique is the quasireversibility method. An approximate global convergence theorem is proved. Results of two numerical experiments are presented. Bibliography: 46 titles. Illustrations: 2 figures. [ABSTRACT FROM AUTHOR]

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex ( E, d) of 'intrinsic' differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials d appear naturally as limits of usual 'Riemannian' differentials d, ε > 0. More precisely, we show that L-energies associated with ε d on 1 forms Γ-converge, as ε → 0, to the energy associated with d. [ABSTRACT FROM AUTHOR]

A new nonlinear spectral conjugate descent method for solving unconstrained optimization problems is proposed on the basis of the CD method and the spectral conjugate gradient method. For any line search, the new method satisfies the sufficient descent condition gkTdk < -‖gk‖ . Moreover, we prove that the new method is globally convergent under the strong Wolfe line search. The numerical results show that the new method is more effective for the given test problems from the CUTE test problem library (Bongartz et al., 1995) in contrast to the famous CD method, FR method, and PRP method. [ABSTRACT FROM AUTHOR]

Abstract: We present a numerical analysis to solve a parameter identification problem. We identify the demographical parameters of a multistage population dynamics model (Ainseba et al., 2011 ). Our nonlinear optimization problem with constraints is solved by a Quasi-Newton method. The convergence proof of this numerical method is performed here. Some numerical applications of it are also given at the end of the paper. [Copyright &y& Elsevier]

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators. [ABSTRACT FROM AUTHOR]

Abstract: Tempered stable processes are widely used in various fields of application as alternatives with finite second moment and long-range Gaussian behaviors to stable processes. Infinite shot noise series representation is the only exact simulation method for the tempered stable process and has recently attracted attention for simulation use with ever improved computational speed. In this paper, we derive series representations for the tempered stable laws of increasing practical interest through the thinning, rejection, and inverse Lévy measure methods. We make a rigorous comparison among those representations, including the existing one due to Imai and Kawai  and Rosiński (2007) , in terms of the tail mass of Lévy measures which can be simulated under a common finite truncation scheme. The tail mass are derived in closed form for some representations thanks to various structural properties of the tempered stable laws. We prove that the representation via the inverse Lévy measure method achieves a much faster convergence in truncation to the infinite sum than all the other representations. Numerical results are presented to support our theoretical analysis. [Copyright &y& Elsevier]

Abstract: The concept of almost convergence was introduced by Lorentz in 1948 [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190], and has various applications. In this work we apply this method to prove some Korovkin type approximation theorems. [Copyright &y& Elsevier]

Abstract: We study a finite element scheme for the 3D Navier–Stokes equations (NSE) that globally conserves energy and helicity and, through the use of Scott–Vogelius elements, enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also show the method can be efficiently computed by exploiting a connection between this method, its associated penalty method, and the method arising from using grad-div stabilized Taylor–Hood elements. Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme. [Copyright &y& Elsevier]

Abstract: New numerical techniques are presented for the solution of a two-dimensional anomalous sub-diffusion equation with time fractional derivative. In these methods, standard central difference approximation is used for the spatial discretization, and, for the time stepping, two new alternating direction implicit (ADI) schemes based on the L 1 approximation and backward Euler method are considered. The two ADI schemes are constructed by adding two different small terms, which are different from standard ADI methods. The solvability, unconditional stability and H 1 norm convergence are proved. Numerical results are presented to support our theoretical analysis and indicate the efficiency of both methods. [Copyright &y& Elsevier]

Abstract: Motivated by the idea of imposing paralleling computing on solving stochastic differential equations (SDEs), we introduce a new domain decomposition scheme to solve forward–backward stochastic differential equations (FBSDEs) parallel. We reconstruct the four step scheme in Ma et al. (1994) and then associate it with the idea of domain decomposition methods. We also introduce a new technique to prove the convergence of domain decomposition methods for systems of quasilinear parabolic equations and use it to prove the convergence of our scheme for the FBSDEs. [Copyright &y& Elsevier]

Abstract: We present a numerical study of the NS-αβ model, which is a recently proposed multiscale variation of the NS-α model that attempts to recapture scales lost through over-regularization by separately modeling dissipation-range scales. We develop a similarity theory for the new model which shows that it is better equipped than the NS-α model to capture smaller-scale behavior. Next, we propose and study an unconditionally stable, optimally accurate, and efficient finite-element implementation for the NS-αβ model; rigorous proofs for stability and convergence are provided. Finally, we present results from two numerical experiments that demonstrate the advantages of the NS-αβ model over the NS-α model. [Copyright &y& Elsevier]