Numerical solutions for optimal control problem governed by elliptic system on Lipschitz domains

Figures & data

Table 1. One-dimensional problem with Neumann conditions.

N	α	β	$L^2-\mathrm{error}$		$H^1-\mathrm{error}$
			JGM	LGM [18]	JGM	LGM [18]
	0.5	0.5	$3.63\times {10}^{-5}$		$9.35\times {10}^{-5}$
	0.5	−0.5	$8.59\times {10}^{-5}$		$1.40\times {10}^{-4}$
	0.0.1	0.01	$6.68\times {10}^{-6}$		$5.39\times {10}^{-5}$
4	0	0	$6.68\times {10}^{-6}$	$2.22\times {10}^{-4}$	$5.39\times {10}^{-5}$	$1.88\times {10}^{-3}$
	$-0.5$	0.5	$8.88\times {10}^{-5}$		$1.43\times {10}^{-4}$
	$-0.5$	−0.5	$3.58\times {10}^{-5}$		$9.59\times {10}^{-5}$
	$-0.99$	−0.99	$6.23\times {10}^{-5}$		$1.64\times {10}^{-4}$
	0.5	0.5	$6.19\times {10}^{-10}$		$1.47\times {10}^{-9}$
	0.5	−0.5	$1.39\times {10}^{-9}$		$2.18\times {10}^{-9}$
	0.01	0.01	$\mathrm{4.24.}\times {10}^{-11}$		$5.86\times {10}^{-10}$
8	0	0	$4.15\times {10}^{-11}$	$7.91\times {10}^{-9}$	$5.85\times {10}^{-10}$	$1.34\times {10}^{-7}$
	$-0.5$	0.5	$1.40\times {10}^{-9}$		$2.19\times {10}^{-9}$
	$-0.5$	−0.5	$3.54\times {10}^{-10}$		$1.01\times {10}^{-9}$
	$-0.99$	−0.99	$4.38\times {10}^{-10}$		$1.32\times {10}^{-9}$
	0.5	0.5	$1.02\times {10}^{-16}$		$1.89\times {10}^{-16}$
	0.5	−0.5	$3.14\times {10}^{-16}$		$3.46\times {10}^{-16}$
	0.01	0.01	$1.75\times {10}^{-16}$		$1.92\times {10}^{-16}$
16	0	0	$2.41\times {10}^{-16}$	$3.10\times {10}^{-14}$	$3.16\times {10}^{-16}$	$1.25\times {10}^{-13}$
	$-0.5$	0.5	$1.64\times {10}^{-16}$		$1.85\times {10}^{-16}$
	$-0.5$	−0.5	$2.26\times {10}^{-16}$		$2.51\times {10}^{-16}$
	$-0.99$	−0.99	$5.40\times {10}^{-16}$		$6.00\times {10}^{-16}$

Table 2. $L^{\mathrm{\infty }}$-error of  $u-u_N$  for  $N=8,12,16$.

N	α	β	JGM
	0.5	0.5	$5.87\times {10}^{-4}$
16	0	0	$7.64\times {10}^{-4}$
	−0.5	−0.5	$1.05\times {10}^{-3}$
	0.5	0.5	$2.91\times {10}^{-8}$
24	0	0	$3.95\times {10}^{-8}$
	−0.5	−0.5	$5.18\times {10}^{-8}$
	0.5	0.5	$1.01\times {10}^{-13}$
32	0	0	$1.42\times {10}^{-13}$
	−0.5	−0.5	$1.92\times {10}^{-13}$

Table 3. $L^{\mathrm{\infty }}$-error of  $u-u_N$  for  $N=8,12,16$.

N	α	β	Case 1	Case 2	Case 3	Case 4
	0.75	0.75	$3.66\times {10}^{-6}$	$2.50\times {10}^{-7}$	$7.48\times {10}^{-6}$	$1.49\times {10}^{-5}$
	0.5	0.5	$2.28\times {10}^{-6}$	$2.08\times {10}^{-7}$	$4.67\times {10}^{-6}$	$9.08\times {10}^{-6}$
8	0	0	$4.08\times {10}^{-7}$	$1.88\times {10}^{-7}$	$2.47\times {10}^{-7}$	$2.57\times {10}^{-7}$
	−0.5	−0.5	$1.15\times {10}^{-6}$	$2.90\times {10}^{-7}$	$2.36\times {10}^{-6}$	$5.36\times {10}^{-6}$
	−0.75	−0.75	$1.45\times {10}^{-6}$	$3.70\times {10}^{-7}$	$2.95\times {10}^{-6}$	$6.52\times {10}^{-6}$
	0.75	0.75	$2.00\times {10}^{-10}$	$1.03\times {10}^{-11}$	$4.05\times {10}^{-10}$	$8.49\times {10}^{-10}$
	0.5	0.5	$1.14\times {10}^{-10}$	$7.47\times {10}^{-12}$	$2.32\times {10}^{-10}$	$4.80\times {10}^{-10}$
12	0	0	$1.20\times {10}^{-11}$	$6.20\times {10}^{-12}$	$7.21\times {10}^{-12}$	$7.10\times {10}^{-12}$
	−0.5	−0.5	$4.40\times {10}^{-11}$	$9.10\times {10}^{-12}$	$8.96\times {10}^{-11}$	$2.00\times {10}^{-10}$
	−0.75	−0.75	$4.92\times {10}^{-11}$	$1.11\times {10}^{-11}$	$9.99\times {10}^{-11}$	$2.20\times {10}^{-10}$
	0.75	0.75	$1.10\times {10}^{-14}$	$6.38\times {10}^{-15}$	$9.18\times {10}^{-15}$	$1.64\times {10}^{-14}$
	0.5	0.5	$1.16\times {10}^{-14}$	$6.21\times {10}^{-15}$	$9.16\times {10}^{-15}$	$1.74\times {10}^{-14}$
16	0	0	$9.83\times {10}^{-15}$	$7.10\times {10}^{-15}$	$7.59\times {10}^{-15}$	$1.33\times {10}^{-14}$
	−0.5	−0.5	$1.10\times {10}^{-14}$	$5.32\times {10}^{-15}$	$8.98\times {10}^{-15}$	$2.52\times {10}^{-14}$
	−0.75	−0.75	$1.44\times {10}^{-14}$	$6.62\times {10}^{-15}$	$7.52\times {10}^{-15}$	$5.20\times {10}^{-14}$

Auteri F, Parolini N, Quartapelle L. Essential imposition of Neumann Galerkin-Legendre elliptic solvers. J Comput Phys. 2003;185:427–444. doi: 10.1016/S0021-9991(02)00064-5

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