Fully Discrete Finite Element Approximation for the Stabilized Gauge-Uzawa Method to Solve the Boussinesq Equations

Table 1

Error decay for Algorithm 1 with 𝜏=ℎ and 𝜇=1.

𝜏=ℎ	1/16	1/32	1/64	1/128	1/256
‖𝐸‖0	0.00198001	0.000651734	0.000188113	5.0822𝑒−05	1.32686𝑒−05
	Order	1.603153	1.792684	1.888075	1.937437
‖𝐸‖𝐿^∞	0.00837795	0.00282504	0.000825997	0.000225008	5.90462𝑒−05
	Order	1.568326	1.774063	1.876160	1.930060
‖𝐸‖1	0.0279984	0.00950852	0.00280233	0.00077052	0.000205078
	Order	1.558052	1.762594	1.862723	1.909660
‖𝑒‖0	0.0341734	0.0123334	0.00376493	0.00105695	0.000283961
	Order	1.470303	1.711876	1.832716	1.896142
‖𝑒‖𝐿^∞	0.197857	0.107171	0.0453294	0.0170497	0.00610541
	Order	0.884544	1.241396	1.410701	1.481586
‖𝜗‖0	0.000215482	5.3851𝑒−05	1.34612𝑒−05	3.36511𝑒−06	8.4091𝑒−07
	Order	2.000522	2.000166	2.000081	2.000630
‖𝜗‖𝐿^∞	0.000197687	4.98479𝑒−05	1.25012𝑒−05	3.1314𝑒−06	7.84502𝑒−07
	Order	1.987613	1.995466	1.997187	1.996959
‖𝜗‖1	0.0167834	0.00839145	0.0041957	0.00209784	0.00104892
	Order	1.000043	1.000009	1.000007	1.000000