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For solving large scale linear systems, a fast convergent randomized Kaczmarz-type method was constructed in Bai and Wu (2018) [4]. In this paper, we propose a geometric probability randomized Kaczmarz (GPRK) method by introducing a new index set J k and three supervised probability criteria defined on J k from a geometric point of view. Linear convergence of GPRK is proved, and the way of argument for the analysis of GPRK also leads to new sharper upper bounds for the randomized Kaczmarz (RK) method and the greedy randomized Kaczmarz (GRK) method. In practice, GPRK is implemented with a simple geometric probability criterion, i.e., the most efficient one of the aforementioned three supervised probability criteria defined on J k. The numerical results demonstrate that GPRK is robust and efficient, and it is faster than GRK in most of the tests in the sense of computing time. [ABSTRACT FROM AUTHOR]

Abstract In this paper various hybrid hp -finite element methods are discussed for elliptic model problems. In particular, a stabilized primal-hybrid hp -method is introduced which approximatively ensures continuity conditions across element interfaces and avoids the enrichment of the primal discretization space as usually required to fulfill some discrete inf-sup condition. A priori as well as a posteriori error estimates are derived for this method. The stabilized primal-hybrid hp -method is also applied to a model obstacle problem since it admits the definition of pointwise constraints. The paper also describes some extensions of primal, primal-mixed and dual-mixed methods as well as their hybridizations to hp -finite elements. In numerical experiments the convergence properties of the stabilized primal-hybrid hp -method are discussed and compared to all the other hp -methods introduced in this paper. The applicability of the a posteriori error estimates to drive h - and hp -adaptive schemes is also investigated. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by combining the p -order block boundary value methods with the m -th Lagrange interpolation, a class of new numerical methods for solving nonlinear fractional differential equations with the γ -order (0 < γ < 1) Caputo derivatives are obtained. It is proved under some appropriate conditions that the induced methods are convergent of order min ⁡ { p , m − γ + 1 } and globally stable. Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods. Highlights • This paper deals with numerical computation and analysis for nonlinear fractional differential equations (FDEs) with γ -order Caputo derivatives. • A class of extended block boundary value methods (EBBVMs) for solving the FDEs are obtained. • The EBBVMs are proved to be convergent of order min ⁡ { p , m − γ + 1 } and globally stable under the appropriate conditions. • Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods. [ABSTRACT FROM AUTHOR]

The unconditional convergence of finite element method for two-dimensional time-dependent viscoelastic flow with an Oldroyd B constitutive equation is given in this paper, while all previous works require certain time-step restrictions. The approximation is stabilized by using the Discontinuous Galerkin (DG) approximation for the constitutive equation. The analysis bases on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element approximation of corresponding iterated time-discrete PDEs. The approach used in this paper can be applied to more general couple nonlinear parabolic and hyperbolic systems. [ABSTRACT FROM AUTHOR]

In this paper we study the multigrid methods for adaptively refined finite element meshes. In our multigrid iterations, on each level we only perform relaxation on new nodes and the old nodes whose support of nodal basis function have changed. The convergence analysis of the algorithm is based on the framework of subspace decomposition and subspace correction. In order to decompose the functions from the finest finite element space into each level, a new projection is presented in this paper. Briefly speaking, this new projection can be seemed as the weighted average of the local L 2 projection. We can perform our subspace decomposition through this new projection by its localization property. Other properties of this new projection are also presented and by these properties we prove the uniform convergence of the algorithm in both 2D and 3D. We also present some numerical examples to illustrate our conclusion. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O ( k + h l ) for l = 1 or l = 2 considering either first or second order finite-element approximations ( k and h being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint k ≤ C h 2 . [ABSTRACT FROM AUTHOR]

This paper presents a class of stationary iterative processes with convergence order equal to the growth rate of generalized Fibonacci sequences. We prove that the informational and computational efficiency of the processes of our class tends to 2 from below. The paper illustrates a connection of the methods of the class with the nonstationary iterative method suggested by our previous paper, whose efficiency index equals to 2. We prove that the efficiency of the nonstationary iterative method, measured by Ostrowski–Traub criteria, is maximal among all iterative processes of order 2. [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]

This paper improves the inexact Kleinman–Newton method for solving algebraic Riccati equations by incorporating a line search and by systematically integrating the low-rank structure resulting from ADI methods for the approximate solution of the Lyapunov equation that needs to be solved to compute the Kleinman–Newton step. A convergence result is presented that tailors the convergence proof for general inexact Newton methods to the structure of Riccati equations and avoids positive semi-definiteness assumptions on the Lyapunov equation residual, which in general do not hold for low-rank approaches. In the convergence proof of this paper, the line search is needed to ensure that the Riccati residuals decrease monotonically in norm. In the numerical experiments, the line search can lead to substantial reduction in the overall number of ADI iterations and, therefore, overall computational cost. [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 The numerical solution of the logarithmic Schrödinger equation on unbounded domains is considered in this paper. It is difficult to develop numerical methods for the logarithmic Schrödinger equation on unbounded domains, due to the blow up of the logarithmic nonlinearity and the unboundedness of the physical domain. Thus, a regularized version of the logarithmic Schrödinger equation on unbounded domains with a small regularization parameter is developed. Then, the local artificial boundary conditions for the regularized logarithmic Schrödinger equation are designed by applying the unified approach, which based on the idea of well-known operator splitting method. The regularized logarithmic Schrödinger equation defined on unbounded domains is reduced to an initial boundary value problem on the bounded computational domain, which can be solved by the finite difference method. The convergence and the stability of the reduced problem are analyzed by introducing some auxiliary variables. In order to choose the optimal absorb parameter in the local artificial boundary conditions, an adaptive algorithm is presented. Numerical results are reported to verify the accuracy and effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose a linear, unconditionally energy stable and second-order (in time) numerical scheme based on a convex splitting scheme and the semi-implicit spectral deferred correction (SISDC) method for the phase field crystal equation. The convex splitting scheme we use is linear, uniquely solvable and unconditionally energy stable but is of first-order, so we take the SISDC method to improve the rate of convergence. The resulted scheme inherits the advantages of the convex splitting scheme and thus leads to linear equations at each time step, which is easy to implement. We also prove that the scheme is unconditionally weak energy stable and of second-order accuracy in time. Numerical experiments are presented to validate the accuracy, efficiency and energy stability of the proposed numerical strategy. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a two-level space–time hybrid Schwarz preconditioner for GMRES based on the classical hybrid Schwarz method and the second-order backward differentiation formula. In the proposed method, the parabolic equations are solved in parallel on both of the space and time directions. Under some reasonable assumptions, the optimal convergence theory is developed for the proposed space–time method, i.e., the convergence rate is independent of the mesh parameters, the number of subdomains and the window size. Some numerical results are given to confirm the theory very well in terms of the convergence rate and accuracy. And the strong/weak scalability obtained with 4096 processors is also reported to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract The paper presents two Tailored Finite Point method (TFPM) for the diffusion equations, which is valid for the strongly anisotropic tensor diffusivity and interface layers. The first scheme uses the value as well as their derivatives at the grid points to construct the five point scheme for the heterogeneous rotating anisotropy. The second scheme is based on the interface conditions to construct the four point scheme for each cell, which gives rise to desirable internal layers and discontinuous diffusivity conditions. Numerically, both methods can achieve uniform convergence, even when there exhibit interface layers. Numerical experiments are presented to show the performance of the proposed two different schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O (τ 2 + h 4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L 2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H 1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a two-grid mixed finite element scheme for distributed optimal control problems governed by stationary Stokes equations. In order to avoid the difficulty caused by the symmetry constraint of the stress tensor, we use pseudostress to replace it. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We first prove that the difference between the interpolation and the numerical solution has superconvergence property for the control u with order h 2. Then, using the postprocessing technique, we derive a second-order superconvergent result for the control u . Next, we construct a two-grid mixed finite element scheme and derive a priori error estimates. Finally, a numerical experiment is presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by further study on the generating function of the discretization matrix, second order convergence of the scheme is proved for diffusion coefficients satisfying a certain condition but are not necessary to be proportional. The theoretical results are justified by numerical tests. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L 2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p + 1) -th order convergent in the L 2 -norm, when piecewise polynomials of degree at most p are used. We further prove that the p -degree DG solution and its derivatives up to order three are O (h 2 p − 2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for P p polynomials with degree p ≥ 3 , and for the classical boundary conditions. [ABSTRACT FROM AUTHOR]

Highlights • An ADI spectral method for the 2D multi-term time fractional anomalous equation is proposed. • The stability and convergence theorems of the ADI spectral method are proved. • The non-smooth solution case for this equation is also considered. • We add some correction terms to the ADI spectral method to achieve high accuracy. Abstract In this paper, we consider the initial boundary value problem of the two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations. An alternating direction implicit (ADI) spectral method is developed based on Legendre spectral approximation in space and finite difference discretization in time. Numerical stability and convergence of the schemes are proved, the optimal error is O (N − r + τ 2) , where N , τ , r are the polynomial degree, time step size and the regularity of the exact solution, respectively. We also consider the non-smooth solution case by adding some correction terms. Numerical experiments are presented to confirm our theoretical analysis. These techniques can be used to model diffusion and transport of viscoelastic non-Newtonian fluids. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a finite element approximation for a linear, first-order in time, unconditionally energy stable scheme proposed in [7] for solving the magneto-hydrodynamic equations. We first reformulate the semi-discrete scheme to the fully discrete version and then carry out a rigorous stability and error analysis for it. We show that the fully discrete scheme indeed leads to optimal error estimates for both velocity and magnetic field with some reasonable regularity assumptions. Moreover, under an alleviated time step constraint (δ t ≤ 1 / | l o g (h) | for 2D and δ t ≤ h for 3D), the optimal error estimate for the pressure is derived as well. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider stochastic Runge–Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for stochastic multi-symplecticity of stochastic Runge–Kutta methods. To present more clearly, we apply these ideas to three dimensional stochastic Maxwell equations driven by multiplicative noise, which play an important role in stochastic electromagnetism and statistical radiophysics areas. Theoretical analysis shows that the methods inherit the energy conservation law of the original system, and preserve the discrete stochastic multi-symplectic conservation law almost surely. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce and analyze a mixed virtual element method (mixed-VEM) for a pseudostress-displacement formulation of the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. We follow a previous work by some of the authors, and employ a mixed formulation that does not require symmetric tensor spaces in the finite element discretization. More precisely, the main unknowns here are given by the pseudostress and the displacement, whereas other physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through simple postprocessing formulae in terms of the pseudostress variable. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, we utilize a well-known local projector onto a suitable polynomial subspace to define a calculable version of our discrete bilinear form, whose continuous version requires information of the variables on the interior of each element. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Babuška–Brezzi theory. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solutions as well as for the fully computable projections of them. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken H (div) -norm. In addition, this postprocessing formula can also be applied to the postprocessed stress tensor. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the backward problem introduced in [48] for Riesz–Feller fractional diffusion. To begin with, some basic properties of solution of the corresponding forward problem, such as the L p estimates, symmetry property and asymptotic estimates, are established by Fourier analysis technique. And then, under various a priori bound assumptions, we give the L 2 conditional stability estimates for the solution of backward problem and also its symmetry property. Moreover, in order to overcome the ill-posedness of the backward problem, we propose a new nonlocal regularization method (NLRM) to solve it. That is, the following nonlocal variational functional is introduced J (φ) = 1 2 ‖ u (φ (x) ; x , T) − f δ (x) ‖ 2 + β 2 ‖ [ P α 1 ⁎ φ ] (x) ‖ 2 , where β ∈ (0 , 1) is a regularization parameter, "⁎" denotes the convolution operation and P α 1 (x) is called a convolution kernel with parameter α 1 , which will be selected properly later. The minimizer of above variational problem is defined as the regularization solution, and the L 2 estimates, symmetry property of regularization solution are given. These results actually show the well-posedness of nonlocal variational problem. Our idea is essentially that using this well-posed problem to approximate the backward (ill-posed) problem. Thus, under an a posteriori parameter choice rule, we deduce various convergence rate estimates under different a-priori bound assumptions for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective and adaptive for different a-priori information. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with nonlinear stochastic functional differential equations with piecewise continuous arguments (SFDEPCAs). Based on an adaptation to the underlying one-leg θ -methods for ODEs, a class of new one-parameter methods for nonlinear SFDEPCAs are introduced. The mean-square exponential stability criteria of analytical and numerical solutions are derived. Under the suitable conditions, it is proved that the one-parameter methods are convergent with strong order 1/2. Some numerical experiments are given to illustrate the theoretical results and computational advantages of the induced methods. Highlights • Stochastic one-leg θ -methods for nonlinear SFDEPCAs are proposed. • The mean-square exponential stability criteria of analytical and numerical solutions are derived. • It is proved under some conditions that stochastic one-leg θ -methods are convergent with strong order 1 2. • The numerical experiment shows that stochastic one-leg θ -methods have the greater computational advantages than stochastic split-step θ -methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second-order time accurate convex splitting scheme for the phase field crystal model. The temporal discretization is based on the second-order backward differentiation formula (BDF) and a convex splitting of the energy functional. The mass conservation, unconditionally unique solvability, unconditionally energy stability and convergence of the numerical scheme are proved rigorously. Mixed finite element method is employed to obtain the fully discrete scheme due to a sixth-order spatial derivative. Numerical experiments are presented to demonstrate the accuracy, mass conservation, energy stability and effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

It is known that the spectral radius of the iterative tensor can be seen as an approximate convergence rate for solving multi-linear systems by tensor splitting iterative methods. So in this paper, first we give some spectral radius comparisons between two different iterative tensors. Then, we propose the preconditioned tensor splitting method for solving multi-linear systems, which provides an alternative algorithm with the choice of a preconditioner. In particular, also we give some spectral radius comparisons between the preconditioned iterative tensor and the original one. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned methods. [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 propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate are established by using a general majorant condition. Two applications of such condition are provided: one is for functions whose derivatives satisfy a Hölder-like condition and the other is for functions that satisfy a Smale condition, which includes a substantial class of analytic functions. Some preliminary numerical experiments illustrating the applicability of the proposed method are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h = H 2 , the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method. [ABSTRACT FROM AUTHOR]

In this paper, a class of modulus-based matrix splitting iteration methods for the quasi-complementarity problems is presented. The convergence analysis of the proposed methods is discussed. Numerical experiments show that the proposed methods are efficient. [ABSTRACT FROM AUTHOR]

In this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs under no regularity assumptions on the solution of the problem. The proposed analysis provides three main contributions to the existing theory. The first novel contribution of this study is a convergence bound that depends on the number of multigrid smoothing iterations. This result is obtained under no regularity assumptions on the solution of the problem. A similar result has been shown in the literature for the cases of full regularity and partial regularity assumptions. Second, our theory applies to local refinement applications with arbitrary level hanging nodes. More specifically, for the smoothing algorithm we provide subspace decompositions that are suitable for applications where the multigrid spaces are defined on finite element grids with arbitrary level hanging nodes. Third, global smoothing is employed on the entire multigrid space with hanging nodes. When hanging nodes are present, existing multigrid strategies advise to carry out the smoothing procedure only on a subspace of the multigrid space that does not contain hanging nodes. However, with such an approach, if the number of smoothing iterations is increased, convergence can improve only up to a saturation value. Global smoothing guarantees an arbitrary improvement in the convergence when the number of smoothing iterations is increased. Numerical results are also included to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, the convergence analysis of operator splitting methods for the Camassa–Holm equation is provided. The analysis is built upon the regularity of the Camassa–Holm equation and the divided equations. It is proved that the solution of the Camassa–Holm equation satisfies the locally Lipschitz condition in H 1 and H 2 norm, which ensures the regularity of the numerical solution. Through the calculus of Lie derivatives, we show that the Lie–Trotter and Strang splitting converge with the expected rate under suitable assumptions. Numerical experiments are presented to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]

This paper is concerned with the adaptation of alternating direction implicit (ADI) time discretization schemes for the numerical solution of partial integro-differential equations (PIDEs) with application to the Bates model in finance. Three different adaptations are formulated and their (von Neumann) stability is analyzed. Ample numerical experiments are provided for the Bates PIDE, illustrating the actual stability and convergence behaviour of the three adaptations. [ABSTRACT FROM AUTHOR]

The partially truncated Euler–Maruyama (EM) method was recently proposed in our earlier paper [3] for highly nonlinear stochastic differential equations (SDEs), where the finite-time strong L r -convergence theory was established. In this note, we will point out that one condition imposed there is restrictive in the sense that this condition might force the stepsize to be so small that the partially truncated EM method would be inapplicable. In this note, we will remove this restrictive condition but still be able to establish the finite-time strong L r -convergence rate. The advantages of our new results will be highlighted by the comparisons with our earlier results in [3] . [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]

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 conservative local discontinuous Galerkin method for a one-dimensional nonlinear Schrödinger equation. By using special generalized alternating numerical fluxes, we establish the optimal rate of convergence O ( h k + 1 ) , with polynomial of degree k and grid size h . Meanwhile, we show that this method preserves the charge conservation law. Numerical experiments verify our theoretical result. [ABSTRACT FROM AUTHOR]

This paper concerns on the introduction of a method for solving a class of Volterra integro-differential equations (VIDEs) of the second kind. It is based on the combination of special general linear methods for ordinary differential equations and Gregory quadrature rule and equipped with a starting procedure. The convergence and stability of the method are analyzed. Some numerical experiments are given to illustrate the agreement of our implementation with the theoretical convergence orders and show the capability of the method in solving nonlinear VIDEs. [ABSTRACT FROM AUTHOR]

In this paper, the superconvergent property is found for the interpolation error of the nonconforming finite element at element centers. Based upon this property, the superconvergence results in the discrete l 2 norm for the solutions E → , H → and c u r l → E → are presented for the 3D time-harmonic Maxwell's equations. In order to get the global superconvergence, a new postprocess operator derived from the rotated Q 1 element interpolation is constructed, which is based on the superconvergence points. All theoretical results are justified by the provided smoothing and non-smoothing numerical tests. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the convergence, properties and error bounds of the discrete solutions of a class of nonlinear systems of reaction–diffusion nonlocal type with moving boundaries, using the finite element method with polynomial approximations of any degree and some classical time integrators. A coordinate transformation which fixes the boundaries is used. Some numerical tests to compare our Matlab code with a moving finite element method are investigated. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [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]

In this paper, we define a new set of functions called fractional-order Bernoulli functions (FBFs) to obtain the numerical solution of linear and nonlinear fractional integro-differential equations. The properties of these functions are employed to construct the operational matrix of the fractional integration. By using this matrix and the least square approximation method the fractional integro-differential equations are reduced to systems of algebraic equations which are solved through the Newton's iterative method. The convergence of the method is extensively discussed and finally, some numerical examples are shown to illustrate the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

We propose and rigorously analyze the subcycling method based on primal domain decomposition techniques for first-order transient partial differential equations. In time dependent problems, it can be computationally advantageous to use different time steps in different regions. Smaller time steps are used in regions of significant changes in the solution and larger time steps are prescribed in regions with nearly stationary response. Subcycling can efficiently reduce the total computational cost. Crucial to our approach is a nonstandard heterogeneous temporal discretization. We begin with the discretization in time by the asynchronous Rothe method, which, in essence, involves a backward finite difference scheme assuming different time steps (fine and large time steps) in different parts of the computational domain. The emphasis of the paper is on qualitative properties of the new numerical scheme, such as a-priori estimates, existence of the time-discrete solutions and the strong convergence and stability analysis. Several numerical experiments were conducted to examine the consistency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we propose a new boundary integral method for the numerical solution of Neumann problems for the Laplace equation, posed in exterior planar domains with piecewise smooth boundaries. Using the single layer representation of the potential, the differential problem is reformulated as a classical boundary integral equation. The use of a smoothing transformation and the introduction of a modified Gauss–Legendre quadrature formula for the approximation of the singular integrals, which turns out to be convergent, leads us to apply a Nyström type method for the numerical solution of the integral equation. We solve some test problems and present the numerical results in order to show the efficiency of the proposed procedure. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of accelerating convergence to equilibrium for the Navier–Stokes equation using time relaxation models. We show that the BDF2 based semidiscrete solution of the regularized scheme converges to the steady-state solution of the continuous Navier–Stokes equations, under appropriate conditions. The proof also shows that time relaxation model can be used to accelerate the convergence with the appropriate choice of the parameters. Numerical experiment is presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, a thorough analysis is given for the order which is observed when integrating evolutionary linear partial differential equations with Lawson methods. The analysis is performed under the general framework of C 0 -semigroups in Banach spaces and hence it can be applied to the numerical time integration of many initial boundary value problems which are described by linear partial differential equations. Conditions of regularity and annihilation at the boundary of these problems are then stated to justify the precise order which is observed, including fractional order of convergence. [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]