Application of Galerkin Weighted Residual Method to 2nd, 3rd and 4th order Sturm-Liouville Problems

The aim of this paper is to compute the eigenvalues for a class of linear Sturm-Liouville problems (SLE) with Dirichlet and mixed boundary conditions applying Galerkin Weighted Residual methods. We use Legendre polynomials over [0,1] as trial functions to approximate the solutions of second, third and fourth order SLE problems. We derive rigorous matrix formulations and special attention is given about how the polynomials satisfy the corresponding homogeneous form of Dirichlet boundary conditions of Sturm-Liouville problems. The obtained approximate eigenvalues are compared with the previous computational studies by various methods available in literature. Keywords: Sturm-Liouville problems, eigenvalue, Legendre polynomials, Galerkin method.

Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm-Liouville problems

The differential quadrature method (DQM) and the Boubaker Polynomials Expansion Scheme (BPES) are applied in order to compute the eigenvalues of some regular fourth-order Sturm-Liouville problems. Generally, these problems include fourth-order ordinary differential equations together with four boundary conditions which are specified at two boundary points. These problems concern mainly applied-physics models like the steady-state Euler-Bernoulli beam equation and mechanicals non-linear systems identification. The approach of directly substituting the boundary conditions into the discrete governing equations is used in order to implement these boundary conditions within DQM calculations. It is demonstrated through numerical examples that accurate results for the first kth eigenvalues of the problem, where k= 1,. 2,. 3,. .... , can be obtained by using minimally 2(k+. 4) mesh points in the computational domain. The results of this work are then compared with some relevant studies. © 2011 Elsevier Inc

Superconvergence and \u3ci\u3ea posteriori\u3c/i\u3e error estimates of a local discontinuous Galerkin method for the fourth-order initial-boundary value problems arising in beam theory

In this paper, we investigate the superconvergence properties and a posteriori error estimates of a local discontinuous Galerkin (LDG) method for solving the one-dimensional linear fourth-order initial-boundary value problems arising in study of transverse vibrations of beams. We present a local error analysis to show that the leading terms of the local spatial discretization errors for the k-degree LDG solution and its spatial derivatives are proportional to (k + 1)-degree Radau polynomials. Thus, the k-degree LDG solution and its derivatives are O(hk+2) superconvergent at the roots of (k + 1)-degree Radau polynomials. Computational results indicate that global superconvergence holds for LDG solutions. We discuss how to apply our superconvergence results to construct efficient and asymptotically exact a posteriori error estimates in regions where solutions are smooth. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement. Our results are valid for arbitrary regular meshes and for Pk polynomials with k ≥ 1, and for various types of boundary conditions

Global convergence of \u3ci\u3ea posteriori\u3c/i\u3e error estimates for a discontinuous Galerkin method for one-dimensional linear hyperbolic problems

In this paper we study the global convergence of the implicit residual-based a posteriori error estimates for a discontinuous Galerkin method applied to one-dimensional linear hyperbolic problems. We apply a new optimal superconvergence result [Y. Yang and C.-W. Shu, SIAM J. Numer. Anal., 50 (2012), pp. 3110-3133] to prove that, for smooth solutions, these error estimates at a fixed time converge to the true spatial errors in the L2 -norm under mesh refinement. The order of convergence is proved to be k + 2, when k-degree piecewise polynomials with k ≥ 1 are used. As a consequence, we prove that the DG method combined with the a posteriori error estimation procedure yields both accurate error estimates and O(hk+2) superconvergent solutions. We perform numerical experiments to demonstrate that the rate of convergence is optimal. We further prove that the global effectivity indices in the L2 -norm converge to unity under mesh refinement. The order of convergence is proved to be 1. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be k + 3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using Pk polynomials with k ≥ 1 and for both the periodic boundary condition and the initial-boundary value problem. Several numerical simulations are performed to validate the theory