Table 7.

Error between the true solution ${\stackrel{\sim }{\mathit{v}}}^{h★}$ and the numerical solution ${\stackrel{\sim }{\mathit{v}}}^h$ of the KKT system for different schemes to precondition the reduced space Hessian. We report the absolute and the relative ℓ²-error between ${\stackrel{\sim }{\mathit{v}}}^{h★}$ and ${\stackrel{\sim }{\mathit{v}}}^h$. We report results for different preconditioners (P_REG, P_2L), different choices for the PDE solver (SL(c) for different CFL numbers c and RK2A(0.2)), and different choices for the method to solve for the action of the inverse of the preconditioner (CHEB(10) and PCG(1E−1)). We solve for ${\stackrel{\sim }{\mathit{v}}}^h$ using a PCG method with a tolerance of 1E−12. We consider the test problem SMOOTH A as in Fig. 2 to set up the problem. We solve the system on a grid of size 256 × 256.

run	PC	PDE solver	PC solver	${\Vert \delta \Vert }_2$	${\Vert \delta \Vert }_{2,\text{rel}}$
#1	PREG	RK2A(0.2)	–	7.49E−13	4.14E−14
#2	P2L	SL(0.2)	CHEB(10)	8.17E−14	4.51E−15
#3		SL(1)	PCG(1E−1)	1.63E−12	9.03E−14
#4		SL(1)	CHEB(10)	1.23E−13	6.77E−15
#5		SL(2)	PCG(1E−1)	1.14E−12	6.30E−14
#6		SL(2)	CHEB(10)	1.89E−13	1.05E−14
#7		SL(5)	PCG(1E−1)	4.49E−12	2.48E−13
#8		SL(5)	CHEB(10)	5.22E−13	2.88E−14
#9		SL(10)	PCG(1E−1)	2.06E−12	1.14E−13
#10		SL(10)	CHEB(10)	2.06E−12	1.14E−13