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Abstract This paper deals with the a priori error analysis for a convection-dominated diffusion 2D problem, when applying the HDG method on a family of anisotropic triangulations. It is known that in this case, boundary or interior layers may appear. Therefore, it is important to resolve these layers in order to recover, if possible, the expected order of approximation. In this work, we extend the use of HDG method on anisotropic meshes. In this context, when the discrete local spaces are polynomials of degree k ≥ 0 , this approach is able to recover an order of convergence k + 1 2 in L 2 for all the variables, under certain assumptions on the stabilization parameter and family of triangulations. Numerical examples confirm our theoretical results. Highlights • We develop an a priori error analysis for a HDG scheme defined on anisotropic meshes that are made of triangles. • We require that the family of triangulations satisfy the maximum angle condition, which is usual in this case. • We include numerical examples that validate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. The present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions. The regularization matrices considered have a structure that makes them easy to apply in iterative methods, including methods based on the Arnoldi process. Numerical examples illustrate the properties and effectiveness of the regularization matrices described. [ABSTRACT FROM AUTHOR]

Abstract: In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the -stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation. [Copyright &y& Elsevier]

In this paper, a new nonconforming immersed finite element (IFE) method on triangular Cartesian meshes is developed for solving planar elasticity interface problems. The proposed IFE method possesses optimal approximation property for both compressible and nearly incompressible problems. Its degree of freedom is much less than those of existing finite element methods for the same problem. Moreover, the method is robust with respect to the shape of the interface and its location relative to the domain and the underlying mesh. Both theory and numerical experiments are presented to demonstrate the effectiveness of the new method. Theoretically, the unisolvent property and the consistency of the IFE space are proved. Experimentally, extensive numerical examples are given to show that the approximation orders in L 2 norm and semi- H 1 norm are optimal under various Lamé parameters settings and different interface geometry configurations. [ABSTRACT FROM AUTHOR]

Solving elliptic equations with interfaces has wide applications in engineering and science. The real world problems are mostly in three dimensions, while an efficient and accurate solver is a challenge. Some existing methods that work well in two dimensions are too complicated to be generalized to three dimensions. Although traditional finite element method using body-fitted grid is well-established, the expensive cost of mesh generation is an issue. In this paper, an efficient non-traditional finite element method with non-body-fitted grids is proposed to solve elliptic interface problems. The special cases when the interface cuts though grid points are handled carefully, rather than perturbing the cutting point away to apply the method for general case. Both Dirichlet and Neumann boundary conditions are considered. Numerical experiments show that this method is approximately second order accurate in the L ∞ norm and L 2 norm for piecewise smooth solutions. The large sparse matrix for our linear system also has nice structure and properties. [ABSTRACT FROM AUTHOR]

Abstract: The linear complementarity problem is to find a vector in satisfying , ,, where and are given. In this paper, we use the fact that solving is equivalent to solving the nonlinear equation where is a function from into itself defined by . We build a sequence of smooth functions which is uniformly convergent to the function . We show that, an approximation of the solution of the (when it exists) is obtained by solving for a parameter large enough. Then we give a globally convergent hybrid algorithm which is based on vector divisions and the secant method for solving . We close our paper with some numerical simulations to illustrate our theoretical results, and to show that this method can solve efficiently large-scale linear complementarity problems. [Copyright &y& Elsevier]

Based on the artificial viscosity approach, a characteristic stabilized finite element method is proposed for approximating solutions to the incompressible Navier–Stokes equations in this paper. The natural combination of characteristic method and subgrid artificial viscosity stabilized finite element method retains the best features of both methods. The stability analysis and error estimates are deduced rigorously. Finally, some numerical experiments are given to demonstrate that this method is highly efficient for the nonstationary Navier–Stokes problems. [ABSTRACT FROM AUTHOR]

We present a theoretical analysis of numerical solutions for nonlinear elliptic problems. Our analysis is based on the abstract approximation theory for branches of nonsingular solutions developed by Brezzi, Rappaz, and Raviart (BRR). In most cases of finite element analysis, the same spaces are used for the domain and range of the nonlinear operator concerning the application of BRR theory. This results in a loss of accuracy in the error estimates while numerical experiments show optimal convergence rates. The main contribution of this paper is a theoretical analysis for the optimal convergence rates. This is achieved by choosing different spaces for the domain and range of the nonlinear operator used in the application of BRR theory. Previously, this idea was used for an analysis of Petrov–Galerkin formulation for the BRR theory. Our analysis is developed for the first-order least-squares finite element methods and the mixed Galerkin methods based on first-order systems of equations. [ABSTRACT FROM AUTHOR]

We present an algorithm for the solution of polynomial equations and secular equations of the form ..., which provides guaranteed approximation of the roots with any desired number of digits. It relies on the combination of two different strategies for dealing with the precision of the floating point computation: the strategy used in the package MPSolve of D. Bini and G. Fiorentino [D.A. Bini, G. Fiorentino, Design, analysis and implementation of a multi-precision polynomial rootfinder, Numer. Algorithms 23 (2000) 127-173] and the strategy used in the package Eigensolve of S. Fortune [S. Fortune, An iterated eigenvalue algorithm for approximating the roots of univariate polynomials, J. Symbolic Comput. 33 (5) (2002) 627-646]. The algorithm is based on the Ehrlich-Aberth (EA) iteration, and on several results introduced in the paper. In particular, we extend the concept and the properties of root-neighborhoods from polynomials to secular functions, provide perturbation results of the roots, obtain an effective stop condition for the EA iteration and guaranteed a posteriori error bounds. We provide an implementation, released in the package MPSolve 3.0, based on the GMP library. From the many numerical experiments it turns out that our code is generally much faster than MPSolve 2.0 and of the package Eigensolve. For certain polynomials, like the Mandelbrot or the partition polynomials the acceleration is dramatic. The algorithm exploits the parallel architecture of the computing platform. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of elliptic interface problems via isoparametric finite element methods. First, a convergence analysis is carried which asserts that optimal rates of convergence are recovered in both the energy and the L 2 –norms. Subsequently, the efficiency of isoparametric finite elements for the problem at hand is also assessed via numerical experiments. Results both with smooth and piecewise regular interfaces are presented and discussed. The numerical predictions corroborate the theoretical results and they also indicate that second-order convergence is achieved in the L ∞ -norm. Comparisons between isoparametric finite elements, Lagrangian finite elements and the Immersed Interface Method are also performed. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the Legendre–Galerkin spectral approximation of elliptic optimal control problems with integral state and control constraints. Thanks to the appropriate base functions of the discrete spaces, the discrete system is with sparse coefficient matrices. We first present the optimality conditions of the control system. Then a priori and a posteriori error estimates both in H 1 and L 2 norms are derived. Some numerical tests indicate that the spectral accuracy can be achieved, and the proposed method is competitive for solving control problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we provide a new modification of the variational iteration method (MVIM) for solving van der Pol equations. The modification couples the classical variational iteration method with He’s polynomials, where the He’s polynomials are applied to the approximate solution and the initial condition to eliminate secular terms. For the large ɛ, the numerical results demonstrate that the modification method get an accurate approximate period than the other presented methods. [Copyright &y& Elsevier]

Abstract: In this paper, the solution of fuzzy linear system (FLS) is investigated based on a -level expansion. To this end, -level of FLS is solved for calculating the core of fuzzy solution and then its spreads are obtained by solving an optimization problem with a special objective function. Using our proposed method, if the computed solution satisfies the FLS, it can be as exact or approximated solution. Finally, the existence of solution of FLS is proved in details and some numerical examples are solved to illustrate the accuracy and capability of the method. [Copyright &y& Elsevier]

Abstract: This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the FVP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to solve the problem of identifying an unknown heat source depending simultaneously on both space and time variables. This problem is transformed into an optimization problem and the uniqueness of minimum element is proved rigorously. Then a variational formulation for solving the optimization problem is given. A conjugate gradient method and a finite difference method are used to solve the variational problem. Some numerical examples are also provided to show the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: Linear discrete ill-posed problems of small to medium size are commonly solved by first computing the singular value decomposition of the matrix and then determining an approximate solution by one of several available numerical methods, such as the truncated singular value decomposition or Tikhonov regularization. The determination of an approximate solution is relatively inexpensive once the singular value decomposition is available. This paper proposes to compute several approximate solutions by standard methods and then extract a new candidate solution from the linear subspace spanned by the available approximate solutions. We also describe how the method may be used for large-scale problems. [Copyright &y& Elsevier]

Abstract: Rough set theory, initiated by Pawlak, is a mathematical tool in dealing with inexact and incomplete information. Various types of uncertainty measure such as accuracy measure, roughness measure, etc, which aim to quantify the imprecision of a rough set caused by its boundary region, have been extensively studied in the existing literatures. However, a few of these uncertainty measures are explored from the viewpoint of formal rough set theory, which, however, help to develop a kind of graded reasoning model in the framework of rough logic. To solve such a problem, a framework of uncertainty measure for formulae in rough logic is presented in this paper. Unlike the existing literatures, we adopt an axiomatic approach to study the uncertainty measure in rough logic, concretely, we define the notion of rough truth degree by some axioms, such a notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true. Then based on this fundamental notion, the notions of rough accuracy degree, roughness degree for any formula, and rough inclusion degree, rough similarity degree between any two formulae are also proposed. In addition, their properties are investigated in detail. These obtained results will be used to develop an approximate reasoning model in the framework of rough logic from the axiomatic viewpoint. [Copyright &y& Elsevier]

Abstract: The classic problem for a logistically evolving single species population being harvested involves three parameters: rate constant, carrying capacity and harvesting rate, which are taken to be positive constants. However, in real world situations, these parameters may vary with time. This paper considers the situation where these vary on a time scale much longer than that intrinsic to the population evolution itself. Application of a multiple time scale approach gives approximate explicit closed form expressions for the changing population, that compare favorably with those generated from numerical solutions. [Copyright &y& Elsevier]

Abstract: In this paper, the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered. The solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented, and the expression of the solution for the optimal approximation problem is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, a closed loop nonlinear optimal control approach is investigated for flexible joint manipulators (FJM). The dynamic load carrying capacity (DLCC) of these manipulators is obtained via this approach. The state-dependent Riccati equation (SDRE) technique is used for solving nonlinear optimal control problem. This method uses special parameterization to develop the nonlinear system to a linear structure having state-dependent coefficient matrices. The Taylor series numerical method is addressed by approximating the solution to the SDRE. Simulations for FJM are provided to illustrate the effectiveness of this approach for designing nonlinear feedback controllers and computing DLCC of these manipulators. Also effects of change in spring factors of manipulator and control parameters on resulted trajectory of manipulator are considered. DLCC is calculated subject to the limits in actuators and tracking accuracy. Finally the results are checked by comparison with experimental results. [Copyright &y& Elsevier]

Abstract: One of the widely used methods for solving a nonlinear system of equations is the quasi-Newton method. The basic idea underlining this type of method is to approximate the solution of Newton’s equation by means of approximating the Jacobian matrix via quasi-Newton update. Application of quasi-Newton methods for large scale problems requires, in principle, vast computational resource to form and store an approximation to the Jacobian matrix of the underlying problem. Hence, this paper proposes an approximation for Newton-step based on the update of approximation requiring a computational effort similar to that of matrix-free settings. It is made possible by approximating the Jacobian into a diagonal matrix using the least-change secant updating strategy, commonly employed in the development of quasi-Newton methods. Under suitable assumptions, local convergence of the proposed method is proved for nonsingular systems. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with Newton’s, Chord Newton’s and Broyden’s methods. [Copyright &y& Elsevier]

Abstract: This paper proposes a new ranking method for fuzzy numbers, which uses a defuzzification of fuzzy numbers and a weighting function. Following Saeidifar and Pasha (2008), first, we define a weighted distance measure on fuzzy numbers, and then, by minimizing this distance, the weighted interval and point approximations of fuzzy numbers are obtained. These indices are applied to rank the fuzzy numbers. This method is new and interesting for ranking fuzzy numbers, and it can be applied for solving and optimizing engineering and economics problems in a fuzzy environment. [Copyright &y& Elsevier]

Abstract: In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential–difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential–difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a). [Copyright &y& Elsevier]

Abstract: In this paper, a high-order exponential (HOE) scheme is developed for the solution of the unsteady one-dimensional convection–diffusion equation. The present scheme uses the fourth-order compact exponential difference formula for the spatial discretization and the Padé approximation for the temporal discretization. The proposed scheme achieves fourth-order accuracy in temporal and spatial variables and is unconditionally stable. Numerical experiments are carried out to demonstrate its accuracy and to compare it with analytic solutions and numerical results established by other methods in the literature. The results show that the present scheme gives highly accurate solutions for all test examples and can get excellent solutions for convection dominated problems. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a robust a posteriori residual error estimator for diffusion–convection–reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in , or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results. [ABSTRACT FROM AUTHOR]