An Extended Galerkin Analysis for Linear Elasticity with Strongly Symmetric Stress Tensor

02/26/2020
by   Qingguo Hong, et al.
0

This paper presents an extended Galerkin analysis for various Galerkin methods of the linear elasticity problem. The analysis is based on a unified Galerkin discretization formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor σ_h, displacement u_h inside each element and the modifications of these two variables σ_h and ǔ_h on elementary boundaries. Motivated by many relevant methods in literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problem and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to infinity, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problem. Two families of inf-sup conditions, one known as H^1-philic and another known as H(div)-phillic, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed formulations and also offer a new and unified analysis for many existing methods in literature as a by-product.

READ FULL TEXT

Authors

page 1

page 2

page 3

page 4

08/22/2019

Extended Galerkin Method

A general framework, known as extended Galerkin method, is presented in ...
01/24/2021

An Extended Galerkin analysis in finite element exterior calculus

For the Hodge–Laplace equation in finite element exterior calculus, we i...
03/03/2022

Symmetric mixed discontinuous Galerkin methods for linear viscoelasticity

We propose and rigorously analyse semi- and fully discrete discontinuous...
02/17/2022

Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems

We use the augmented Lagrangian formalism to derive discontinuous Galerk...
09/30/2020

Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems

In this paper, we present a unified analysis of the superconvergence pro...
11/12/2019

Tutorial on Hybridizable Discontinuous Galerkin (HDG) Formulation for Incompressible Flow Problems

A hybridizable discontinuous Galerkin (HDG) formulation of the linearize...
05/13/2022

Hybridized Discontinuous Galerkin Methods for a Multiple Network Poroelasticity Model with Medical Applications

The quasi-static multiple network poroelastic theory (MPET) model, first...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.