On the Local Discontinuous Galerkin Method for Linear Elasticity

Table 1

Numerical errors of LDG method for several choices of $k$ and $l$. The theoretical convergence rates of ${\parallel \mathbf{u}-{\mathbf{u}}_h\parallel }_0\text{,}$${|\mathbf{u}-{\mathbf{u}}_h|}_{1,h}$, and $\parallel |\mathbf{u}-{\mathbf{u}}_h|\parallel$ are $O(h^{k+\min \hspace{0.17em}\{1,k\}})\text{,}$$O(h^k)$, and $O(h^k)$, respectively.

$k,l$	$h$	$2^{-1}$	$2^{-2}$	$2^{-3}$	$2^{-4}$	$2^{-5}$
$k=1,\hspace{0.17em}\hspace{0.17em}l=1$	$\parallel \mathbf{u}-{\mathbf{u}}_h{\parallel }_0$	0.11271	0.03173	0.00827	0.00205	0.00050
	$|\mathbf{u}-{\mathbf{u}}_h{|}_{1,h}$	1.04389	0.52475	0.26098	0.12990	0.06477
	$|\parallel \mathbf{u}-{\mathbf{u}}_h\parallel |$	0.97179	0.48304	0.23833	0.11806	0.05872
$k=1,\hspace{0.17em}\hspace{0.17em}l=0$	$\parallel \mathbf{u}-{\mathbf{u}}_h{\parallel }_0$	0.09448	0.02371	0.00570	0.00128	0.00026
	$|\mathbf{u}-{\mathbf{u}}_h{|}_{1,h}$	1.03341	0.51585	0.25644	0.12773	0.06373
	$|\parallel \mathbf{u}-{\mathbf{u}}_h\parallel |$	1.04905	0.51847	0.25572	0.12673	0.06305
$k=0,\hspace{0.17em}\hspace{0.17em}l=0$	$\parallel \mathbf{u}-{\mathbf{u}}_h{\parallel }_0$	0.77564	0.70692	0.67276	0.65756	0.65067
	$|\mathbf{u}-{\mathbf{u}}_h{|}_{1,h}$	3.37310	3.37310	3.37310	3.37310	3.37310
	$|\parallel \mathbf{u}-{\mathbf{u}}_h\parallel |$	3.24274	3.22244	3.21906	3.21822	3.21790