On the Local Discontinuous Galerkin Method for Linear Elasticity

Yuncheng Chen,Yifeng Xu,Xuehai Huang,Jianguo Huang

doi:10.1155/2010/759547

Abstract

Following Castillo et al. (2000) and Cockburn (2003), a general framework of constructing discontinuous Galerkin (DG) methods is developed for solving the linear elasticity problem. The numerical traces are determined in view of a discrete stability identity, leading to a class of stable DG methods. A particular method, called the LDG method for linear elasticity, is studied in depth, which can be viewed as an extension of the LDG method discussed by Castillo et al. (2000) and Cockburn (2003). The error bounds inL2-norm,H1-norm, and a certain broken energy norm are obtained. Some numerical results are provided to confirm the convergence theory established.

Highlights

This paper is focused on systematically studying discontinuous Galerkin DG methods for the linear elasticity problem
Since the DG method was first introduced in 1970s, these methods have been applied for solving a variety of mathematical-physical problems including linear and nonlinear hyperbolic problems, Navier-Stokes equations, convection-dominated diffusion problems, and so on
The DG method may be viewed as high-order extensions of the classical finite volume method

Summary

Introduction

This paper is focused on systematically studying discontinuous Galerkin DG methods for the linear elasticity problem. We may apply mixed finite-element methods to solve the linear elasticity system, from which the aforementioned two physical quantities can be obtained simultaneously. Due to the symmetry constraint on the stress tensor, it is extremely difficult to construct stable stress-displacement finite elements. A piecewise quadratic stress space is constructed with 162 degrees of freedom on each tetrahedron cf 9 Another approach in this direction is to use composite elements macroelements , in which the discrete displacement space consists of piecewise polynomials with respect to one triangulation of the domain, while the discrete stress space consists of piecewise polynomials with respect to a more refined triangulation cf 10–13. Regarding the complexity of mixed elements given above, the discontinuous Galerkin method is naturally a suitable alternative for numerically solving linear elasticity problems.

Basic Framework for the DG Method

Numerical Traces and the LDG Method

Error Analysis for the LDG Method

Numerical Results