Tension spline method for the solution of elliptic equations

Figures & data

Table 1. The maximum absolute errors and orders of convergence of Example 6.1.

(h,k)	Method ($T_1$)	Conv. order	Method ($T_2$)	Conv. order
(1/16, 1/16)	2.4162 E−3		6.6144 E−6
(1/16, 1/32)	2.0264 E−3		5.6684 E−6
(1/32, 1/16)	9.9654 E−4		1.3588 E−6
(1/32, 1/32)	6.0505 E−4	1.99	4.1319 E−7	4.00
(1/64, 1/64)	1.4752 E−4	2.04	2.5170 E−8	4.04
(1/128, 1/128)	2.3373 E−5	2.66	9.8365 E−10	4.67

Table 2. The maximum absolute errors in Example 6.2.

(h,k)	Our method ($T_2$)	Conv. order	Method [13]	Conv. order
(1/6, 1/6)	2.315 E−5	–	5.72 E−3	–
(1/9, 1/9)	4.758 E−6	3.90	8.29 E−4	4.11
(1/17, 1/17)	3.714 E−7	4.01	5.89 E−5	3.81
(1/21, 1/21)	1.525 E−7	4.21	5.89 E−5	4.01

Table 3. The maximum absolute errors and orders of convergence for Example 6.3.

(h,k)	Method ($T_1$) c=4	Conv. order	Method ($T_2$) c=4	Conv. order	Method ($T_2$) c=16	Conv. order
(0.2, 0.2)	1.0436 E−1		9.5330 E−3		1.8696 E−1
(0.1, 0.1)	2.7798 E−2	1.91	5.6792 E−4	4.07	9.5330 E−3	4.30
(0.05, 0.05)	7.7792 E−3	1.84	3.8782 E−5	3.87	5.6792 E−4	4.06
(0.025,0.025)	1.9548 E−3	1.99	2.4167 E−6	4.00	3.8782 E−5	3.88
(0.02,0.02)	1.2470 E−3	2.02	6.5835 E−7	4.49	1.5800 E−5	4.02

Figure 1. Comparison of exact and approximate solutions for Example 6.1 with $(h,k)=(1/20,1/20)$.

Houstis E, Vavalis E, Rice J. Convergence of o(h4) cubic spline collocation methods for elliptic partial differential equations. SIAM J Numer Anal. 1988;25(1):54–74. doi: 10.1137/0725006

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