An interpolating element-free Galerkin scaled boundary method applied to structural dynamic analysis

An efficient semi-analytical method, namely the interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is developed for structural dynamic analysis in this paper, which is based on boundary scattered nodes with no need of element connectivity. Since the shape functions of the improved interpolating moving least-squares (IIMLS) method satisfy the delta function property, the essential boundary conditions, as a result, can be enforced accurately without any additional efforts. Based on the improved continued fraction, the dynamic properties of a bounded domain are expressed by the high order static stiffness and mass matrices. This continued fraction solution is computationally more robust and the transient response can be obtained directly in the time domain using standard procedures in structural dynamics. It is testified from the computational results that the present method for structural dynamic analysis is quite easy to be implemented with high accuracy.     

Numerical analysis of a continuous Galerkin method for damped sine-Gordon equation

In this article, we discuss the numerical solution for the two-dimensional (2-D) damped sine-Gordon equation by using a space–time continuous Galerkin method. This method allows variable time steps and space mesh structures and its discrete scheme has good stability which are necessary for adaptive computations on unstructured grids. Meanwhile, it can easily get the higher-order accuracy in both space and time directions. The existence and uniqueness to the numerical solution are strictly proved and a priori error estimate in maximum-norm is given without any space–time grid conditions attached. Also, we prove that if the mesh in each time level is generated in a reasonable way, we can get the optimal order of convergence in both temporal and spatial variables. Finally, the convergence rates are presented and analyzed by some numerical experiments to illustrate the validity of the scheme.     

Error estimates for the Galerkin method as applied to time-dependent equations

A projection method is studied as applied to the Cauchy problem for an operator-differential equation with a non-self-adjoint operator. The operator is assumed to be sufficiently smooth. The linear spans of eigenelements of a self-adjoint operator are used as projection subspaces. New asymptotic estimates for the convergence rate of approximate solutions and their derivatives are obtained. The method is applied to initial-boundary value problems for parabolic equations.     

Approximate analytical solutions of Schnakenberg systems by homotopy analysis method

In this paper, the homotopy analysis method (HAM) has been employed to obtain analytical solution of a two reaction–diffusion systems of fractional order (fractional Schnakenberg systems) which has been modeling morphogen systems in developmental biology. Different from all other analytic methods, HAM provides us with a simple way to adjust and control the convergence region of solution series by choosing proper values for auxiliary parameter h. The fractional derivative is described in the Caputo sense. The reason of using fractional order differential equations (FOD) is that FOD are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived of the fractional system are of a more general nature. Respectively, solutions of FOD spread at a faster rate than the classical differential equations, and may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.     

Approximate analysis of production systems

In this paper complex production systems are studied where a single product is manufactured and where each production unit stores its output in at most one buffer and receives its input from at most one buffer. The production units and the buffers may be connected nearly arbitrarily. The buffers are supposed to be of finite capacity and the goods flow is continuous. For such netwroks it is possible to estimate the throughput by applying repeated aggregation over the production units. The approximation appears to be best when the network shows some resemblance with a flow line.     

On the energetic Galerkin boundary element method applied to interior wave propagation problems

We consider two-dimensional interior wave propagation problems with vanishing initial and mixed boundary conditions, reformulated as a system of two boundary integral equations with retarded potential. These latter are then set in a weak form, based on a natural energy identity satisfied by the solution of the differential problem, and discretized by the related energetic Galerkin boundary element method. Numerical results are presented and discussed.     

Galerkin method for Wiener-Hopf operators with piecewise continuous symbol

This paper is concerned with the applicability of maximum defect polynomial (Galerkin) spline approximation methods with graded meshes to Wiener-Hopf operators with matrix-valued piecewise continuous generating function defined on R. For this, an algebra of sequences is introduced, which contains the approximating sequences we are interested in. There is a direct relationship between the stability of the approximation method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras are used in order to derive stability criteria for the approximation sequences.Supported by grant Praxis XXI/BD/4501/94 from FCT.Partly supported by FCT/BMFT grant 423.     

Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems

Summary The purpose of this article is to obtainL ² and uniform norm error estimates for the Galerkin approximation of the solution of certain boundary value problems via a comparison with then-norm projection of the solution. In some cases, these estimates constitute an improvement over known results.This research was supported in part by AEC Grant (11-1)-2075.     

A review of the Local Discontinuous Galerkin (LDG) method applied to elliptic problems

This paper presents a review of the so-called Local Discontinuous Galerkin (LDG) method applied to elliptic problems. The method is presented using a mixed formulation similar to that of the classical mixed finite element method. A summary of the convergence properties is presented. Preliminary theoretical results on super-convergent points are discussed. Numerical experiments of a gradient recovering technique are presented.     

Homotopy analysis method applied to electrohydrodynamic flow

In this paper, we consider the nonlinear boundary value problem (BVP) for the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylindrical conduit. We present analytical solutions based on the homotopy analysis method (HAM) for various values of the relevant parameters and discuss the convergence of these solutions. We also compare our results with numerical solutions. The results provide another example of a highly nonlinear problem in which HAM is the only known analytical method that yields convergent solutions for all values of the relevant parameters.     

Error estimates for a discontinuous Galerkin method with interior penalties applied to nonlinear Sobolev equations

A Discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi‐discrete and a family of fully‐discrete time approximate schemes are formulated. These schemes are symmetric. Hp‐version error estimates are analyzed for these schemes. For the semi‐discrete time scheme a priori L^∞(H¹) error estimate is derived and similarly, l^∞(H¹) and l²(H¹) for the fully‐discrete time schemes. These results indicate that spatial rates in H¹ and time truncation errors in L² are optimal. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations

A discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi-discrete and a family of Fully-discrete time approximate scheme are formulated. These schemes can be symmetric or nonsymmetric. Hp-version error estimates are analyzed for these schemes. Just because of a damping term ·(b(u)u_t) included in Sobolev equation, which is the distinct character different from parabolic equation, special test functions are chosen to deal with this term. Finally, a priori L^∞(H¹) error estimate is derived for the semi-discrete time scheme and similarly, l^∞(H¹) and l²(H¹) for the Fully-discrete time schemes. These results also indicate that spatial rates in H¹ and time truncation errors in L² are optimal.     

An augmented Galerkin procedure for the boundary integral method applied to mixed boundary value problems

Here we present the asymptotic error analysis for the boundary element approximation of the direct boundary integral equations for the plane mixed boundary value problem of the Laplacian. The boundary elements are defined by B-splines for the smooth parts of the boundary charges and additional singular functions at the collision points. The asymptotic error estimates include estimates for the stress intensity factors which occur as additional unknowns to be computed within the Galerkin scheme. The numerical analysis is based on the uniqueness of the problem, a coerciveness inequality, the triangular principal part and an extended shift theorem of the boundary integral operators.     

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.

     

Approximate Bayes estimators applied to the Bilal model

This paper develops approximate Bayes estimators of the parameter of the Bilal failure model by using the method of Tierney and Kadane [Accurate approximations for posterior moments and marginal densities, J. Amer. Statist. Assoc. 81 (1986) 82–86.] based on Type-2 censored sample and four different loss functions. Existence and uniqueness theorem for the maximum likelihood estimate are established. Based on Monte Carlo simulation study, comparisons are made between those estimators and their corresponding Bayes estimators obtained by using Gibb’s sampling approach.     

Improved error estimates for the perturbed Galerkin method applied to a class of generalized eigenvalue problems

We consider the generalized eigenvalue problem x-Kx = μBx in a complex Banach space E. Here, K and B are bounded linear operators, B is compact, and 1 is not in the spectrum of K. If {E_n: n = 1, 2,…} is a sequence of closed subspaces of E and P_n: E → E_n is a linear projection which maps E onto E_n, then we consider the sequence of approximate eigenvalue problems {x_n - P_nKx_n = μP_nBx_n in E_n: n = 1, 2,…}. Assuming that ∥K-P_nK∥ → 0 and t|B-P_nB∥ → 0 as n → ∞, we prove the convergence of sequences of eigenvalues and eigenelements of the approximate eigenvalue problem to eigenvalues and eigenelements of the original eigenvalue problem, and establish upper bounds for the errors. These error bounds are sharper than those given by Vainikko in Ref. 2 for the more general problem x = μTx in E, T linear and compact, and the sequence of approximate problems {x_n = μT_nx_n in E_n: n = 1, 2,…}, and do not involve the operator S_n = T_n-P_nT ∥;E_n.     

Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations

We consider space-time continuous Galerkin methods with mesh modification in time for semilinear second order hyperbolic equations. We show a priori estimates in the energy norm without mesh conditions. Under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates. Estimates of the jump in the Riesz projection in two successive time steps are also derived.