scholarly journals A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xuehai Huang
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Local Discontinuous Galerkin Method ◽  
Element Space ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Nearly Incompressible Linear Elasticity

A reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity is proposed in this paper, which is locking-free. RLDG method can be formally regarded as a special case of LDG method withC11=0. However, RLDG method is actually not covered by LDG method, whereC11must be chosen to be positive to ensure the stability of LDG method. RLDG method can also be considered as the localization of some symmetric nonconforming mixed finite element method. The implementation of RLDG method is discussed. By introducing a lifting operator as LDG method, RLDG method can be rewritten as primal formulation with unknown displacement only. Next, we obtain that the convergence rates of the approximation to stress tensor in energy norm and displacement inL2-norm areO(hk)andO(hk+1), respectively, which are both uniform with respect toλ. Moreover, we obtain aH(div)-conforming displacement by projecting the displacement and corresponding numerical trace of RLDG method into the Raviart-Thomas element space. And then we analyze the error estimates of this postprocessed displacement inH(div)-seminorm andL2-norm, which are also uniform with respect toλ. Finally, some numerical results are shown to demonstrate the theoretical results.

Local Discontinuous Galerkin Method for Time-Dependent Singularly Perturbed Semilinear Reaction-Diffusion Problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yao Cheng ◽  
Chuanjing Song ◽  
Yanjie Mei
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Domain Boundary ◽  
Local Discontinuous Galerkin Method ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Element Space ◽  
Local Discontinuous Galerkin ◽  
Ldg Method

AbstractLocal discontinuous Galerkin method is considered for time-dependent singularly perturbed semilinear problems with boundary layer. The method is equipped with a general numerical flux including two kinds of independent parameters. By virtue of the weighted estimates and suitably designed global projections, we establish optimal {(k+1)}-th error estimate in a local region at a distance of {\mathcal{O}(h\log(\frac{1}{h}))} from domain boundary. Here k is the degree of piecewise polynomials in the discontinuous finite element space and h is the maximum mesh size. Both semi-discrete LDG method and fully discrete LDG method with a third-order explicit Runge–Kutta time-marching are considered. Numerical experiments support our theoretical results.

scholarly journals On the Local Discontinuous Galerkin Method for Linear Elasticity

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Yuncheng Chen ◽  
Jianguo Huang ◽  
Xuehai Huang ◽  
Yifeng Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Linear Elasticity ◽  
General Framework ◽  
Local Discontinuous Galerkin Method ◽  
Convergence Theory ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Dg Methods ◽  
Discrete Stability

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.

scholarly journals Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

2019 ◽  
Vol 53 (6) ◽  
pp. 2081-2108
Author(s):  
Eldar Khattatov ◽  
Ivan Yotov
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Normal Stress ◽  
Linear Elasticity ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Decomposition Methods ◽  
Interface Problem ◽  
Element Formulation ◽  
Element Space

Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.

A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

2014 ◽  
Vol 15 (4) ◽  
pp. 1091-1107 ◽  
Author(s):  
Yinhua Xia ◽  
Yan Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Time Discretization ◽  
Local Discontinuous Galerkin Method ◽  
Implicit Time ◽  
Discrete Version ◽  
Time Step ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Spatial Derivatives ◽  
Number Of Particles

AbstractIn this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

A NEW NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

2006 ◽  
Vol 16 (07) ◽  
pp. 979-999 ◽  
Author(s):  
SON-YOUNG YI
Keyword(s):  
Finite Element ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Displacement Boundary Condition ◽  
Displacement Boundary ◽  
Optimal Convergence Rates ◽  
Optimal Order Convergence

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of [Formula: see text] for the L2-error of the stress and [Formula: see text] for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.

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