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Abstract: The paper deals with the initial–boundary value problem for the linear wave equation where , are given constants and , , are given functions, and the unknown function and the unknown boundary value satisfy the following nonlinear integral equation: where , , , are given constants and , are given functions. In this paper, we consider three main parts. In Part 1 we prove a theorem of global existence and uniqueness of a weak solution of problem (1.1)–(1.5). The proof is based on a Galerkin method associated with a priori estimates, weak convergence and compactness techniques. For the case of , Part 2 is devoted the study of the asymptotic behavior of the solution as . Finally, in Part 3 we obtain an asymptotic expansion of the solution of the problem (1.1)–(1.5) up to order in four small parameters , , , . [Copyright &y& Elsevier]

In this paper, multiscale Galerkin method is presented to approximate the solutions of second-order boundary value problems of Fredholm integro-differential equation. The method is based on traditional Galerkin method and uses the multiscale orthonormal bases to discretize the equations. The proposed method is proved to be stable and have the optimal convergence order. Numerical examples are presented to confirm the theoretical results and show that the method is computationally stable, valid and accurate. [ABSTRACT FROM AUTHOR]

In this article, we develop the Sinc-Galerkin method based on double exponential transformation for solving a class of weakly singular nonlinear two-point boundary value problems with nonhomogeneous boundary conditions. Also several examples are solved to show the accuracy efficiency of the presented method. We compare the obtained numerical results with results of the other existing methods in the literature. The results of this paper confirm that our method is very fast, simple and considerably accurate. [ABSTRACT FROM AUTHOR]

Abstract: The scaled boundary method is a semi-analytical method of analysis which can be used in computational mechanics. This method uses a normalised radial coordinate system, introducing shape functions to weaken the governing equations in the circumferential direction and solving the resulting differential equations analytically in the radial direction. This paper presents a new Element-free Galerkin (EFG) scaled boundary method in which the EFG approach is used in the circumferential direction. The proposed model is verified by application to a number of standard problems of elasticity. The numerical solutions show that the new method has higher accuracy (for any particular number of nodes) and better convergence than scaled boundary finite element methods, and an accurate smooth stress field can be obtained directly without the necessity of using a stress recovery procedure. [Copyright &y& Elsevier]

In this paper, we propose a new discontinuous finite element method to solve initial value problems for ordinary differential equations and prove that the finite element solution exhibits an optimal O(Δt p+1) convergence rate in the norm. We further show that the p-degree discontinuous solution of differential equation of order m and its first m −1 derivatives are O(Δt 2p+2−m ) superconvergent at the end of each step. We also establish that the p-degree discontinuous solution is O(Δt p+2) superconvergent at the roots of (p +1− m)-degree Jacobi polynomial on each step. Finally, we present several computational examples to validate our theory and construct asymptotically correct a posteriori error estimates. [Copyright &y& Elsevier]

Abstract: In this paper, we develop discontinuous Galerkin (DG) methods based on non-polynomial approximation spaces for numerically solving time dependent hyperbolic and parabolic and steady state hyperbolic and elliptic partial differential equations (PDEs). The algorithm is based on approximation spaces consisting of non-polynomial elementary functions such as exponential functions, trigonometric functions, etc., with the objective of obtaining better approximations for specific types of PDEs and initial and boundary conditions. It is shown that L 2 stability and error estimates can be obtained when the approximation space is suitably selected. It is also shown with numerical examples that a careful selection of the approximation space to fit individual PDE and initial and boundary conditions often provides more accurate results than the DG methods based on the polynomial approximation spaces of the same order of accuracy. [Copyright &y& Elsevier]

The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution. In this paper, we prove general order a priori error estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution. [ABSTRACT FROM AUTHOR]

This paper discovers and researches problems on numerical oscillations of the solution in element-free Galerkin method (EFGM) when it uses high order polynomial basis, and puts forward the meshless method based on orthogonal basis (MLMBOB), which is composed of essential boundary conditions with Penalty method, then gets the numerical solutions of the partial differential equations. This method holds nearly all qualities of EFGM and removes many drawbacks of it, and it has high accuracy when high order orthogonal basis is used. Therefore, it is fit for many problems in engineering computational electromagnetics. Examples are given to prove the proposed method. [ABSTRACT FROM AUTHOR]

The paper considers the problem of justifying the asymptotic solution of the phase-field system. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to highlight that, when dealing with DG methods for linear hyperbolic equations or advection-dominated equations, it is much more convenient to write the upwind numerical flux as the sum of the usual (symmetric) average and a jump penalty. The equivalence of the two ways of writing is certainly well known (see e.g. Ref. 4); yet, it is very widespread not to consider upwinding, for DG methods, as a stabilization procedure, and too often in the literature the upwind form is preferred in proofs. Here, we wish to underline the fact that the combined use of the formalism of Ref. 3 and the jump formulation of upwind terms has several advantages. One of them is, in general, to provide a simpler and more elegant way of proving stability. The second advantage is that the calibration of the penalty parameter to be used in the jump term is left to the user (who can think of taking advantage of this added freedom), and the third is that, if a diffusive term is present, the two jump stabilizations (for the generalized upwinding and for the DG treatment of the diffusive term) are often of identical or very similar form, and this can also be turned to the user's advantage. [ABSTRACT FROM AUTHOR]

One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary Γu, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin-based meshless method, such as element-free Galerkin methods, reproducing kernel particle method, meshless local Petrov–Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]