Null Field and Interior Field Methods for Laplace’s Equation in Actually Punctured Disks

Table 1

The errors and condition numbers by the conservative schemes of the IFM, where 𝑅=2.5 and 𝛿=𝑢−𝑢_𝑀−𝑁.

𝑅1	(𝑀,𝑁)	‖𝛿‖∞,𝜕𝑆	‖𝛿‖0,𝜕𝑆	Cond	Cond_eff
1	(24,  12)	1.52 (−8)	1.54 (−8)	8.48 (1)	2.70 (1)
0.5	(24,  12)	4.90 (−11)	4.70 (−11)	2.64 (2)	1.30 (2)
0.1	(24, 8)	6.81 (−12)	6.67 (−12)	1.62 (3)	9.21 (2)
0.1	(24, 7)	6.81 (−12)	6.67 (−12)	1.39 (3)	7.89 (2)
0.1	(24,  6)	6.81 (−12)	6.83 (−12)	1.16 (3)	6.58 (2)
0.1	(24, 5)	2.40 (−10)	9.34 (−11)	9.34 (2)	5.26 (2)
0.1	(24, 4)	1.58 (−8)	6.10 (−9)	7.05 (2)	3.95 (2)
10^−2	(24, 6)	3.69 (−12)	3.62 (−12)	1.97 (4)	1.16 (4)
10^−2	(24, 5)	6.69 (−12)	3.62 (−12)	1.57 (4)	9.26 (3)
10^−2	(24,  4)	3.69 (−12)	3.62 (−12)	1.18 (4)	6.94 (3)
10^−2	(24, 3)	6.10 (−10)	7.63 (−11)	7.91 (3)	4.63 (3)
10^−2	(24, 2)	4.80 (−7)	6.00 (−8)	3.98 (3)	2.31 (3)
10^−3	(24, 5)	2.58 (−12)	2.53 (−12)	2.22 (5)	1.32 (5)
10^−3	(24, 4)	2.58 (−12)	2.53 (−12)	1.67 (5)	9.93 (4)
10^−3	(24,  3)	2.58 (−12)	2.53 (−12)	1.11 (5)	6.62 (4)
10^−3	(24, 2)	3.35 (−9)	1.33 (−10)	5.58 (4)	3.31 (4)
10^−3	(24, 1)	3.52 (−5)	1.39 (−6)	1.09 (2)	7.90 (1)
10^−4	(24, 5)	1.98 (−12)	1.94 (−12)	2.87 (6)	1.72 (6)
10^−4	(24, 4)	1.98 (−12)	1.94 (−12)	2.15 (6)	1.29 (6)
10^−4	(24,  3)	1.98 (−12)	1.95 (−12)	1.44 (6)	8.62 (5)
10^−4	(24, 2)	2.57 (−11)	1.97 (−12)	7.20 (5)	4.31 (5)
10^−4	(24, 1)	2.70 (−6)	3.38 (−8)	1.40 (2)	1.03 (2)