A Proper-Orthogonal Decomposition Variational Multiscale Approximation Method for a Generalized Oseen Problem

Table 2

Comparison of the error between the ROM-calculated velocity 𝐮^𝑟 and the true solution 𝐮 in Experiment 2 for the POD-Galerkin approximation and various choices of 𝑅,𝛼 in the POD-VMS model.

𝑟	𝑅	𝛼	‖𝐮−𝐮^𝑟‖0,2	‖𝐮−𝐮^𝑟‖1,2	Time (s)
10	5	ℎ	6.442288⋅10^−3	2.752475⋅10^−1	933.6
10	8	ℎ	3.882789⋅10^−3	1.101335⋅10^−1	887.1
10	5	4ℎ	8.374255⋅10^−3	3.767016⋅10^−1	911.2
10	8	4ℎ	3.882517⋅10^−3	1.092202⋅10^−1	863.5
10	5	ℎ/4	4.481848⋅10^−3	1.581403⋅10^−1	916.0
10	8	ℎ/4	3.883223⋅10^−3	1.127396⋅10^−1	872.0
10	PODG	N/A	3.883386⋅10^−3	1.137251⋅10^−1	793.1
15	5	ℎ	6.291607⋅10^−3	2.718394⋅10^−1	1397.7
15	10	ℎ	3.882608⋅10^−3	1.136891⋅10^−1	1238.9
15	5	4ℎ	8.266850⋅10^−3	3.738084⋅10^−1	1393.4
15	10	4ℎ	3.883011⋅10^−3	1.136693⋅10^−1	1238.3
15	5	ℎ/4	4.452304⋅10^−3	1.593695⋅10^−1	1388.5
15	10	ℎ/4	3.882695⋅10^−3	1.142659⋅10^−1	1234.5
15	PODG	N/A	3.883556⋅10^−3	1.154404⋅10^−1	1125.1
20	5	ℎ	6.093426⋅10^−3	2.641301⋅10^−1	2130.3
20	10	ℎ	3.882569⋅10^−3	1.136950⋅10^−1	1864.1
20	15	ℎ	3.883766⋅10^−3	1.153448⋅10^−1	1704.7
20	5	4ℎ	8.197183⋅10^−3	3.713549⋅10^−1	2134.3
20	10	4ℎ	3.882755⋅10^−3	1.136407⋅10^−1	1869.3
20	15	4ℎ	3.883493⋅10^−3	1.163518⋅10^−1	1709.3
20	5	ℎ/4	4.350461⋅10^−3	1.521759⋅10^−1	2095.5
20	10	ℎ/4	3.883271⋅10^−3	1.143748⋅10^−1	1847.3
20	15	ℎ/4	3.884202⋅10^−3	1.154097⋅10^−1	1695.4
20	PODG	N/A	3.884723⋅10^−3	1.156279⋅10^−1	1593.2
25	5	ℎ	6.267525⋅10^−3	2.682228⋅10^−1	3250.2
25	10	ℎ	3.882769⋅10^−3	1.137263⋅10^−1	2863.5
25	15	ℎ	3.884018⋅10^−3	1.153711⋅10^−1	2590.2
25	20	ℎ	3.884777⋅10^−3	1.156084⋅10^−1	2426.6
25	5	4ℎ	8.183246⋅10^−3	3.690778⋅10^−1	3212.8
25	10	4ℎ	3.882875⋅10^−3	1.136663⋅10^−1	2829.7
25	15	4ℎ	3.883670⋅10^−3	1.153693⋅10^−1	2549.5
25	20	4ℎ	3.334791⋅10^−3	1.166200⋅10^−1	2391.1
25	5	ℎ/4	4.533282⋅10^−3	1.634145⋅10^−1	3175.2
25	10	ℎ/4	3.883342⋅10^−3	1.143960⋅10^−1	2797.1
25	15	ℎ/4	3.884376⋅10^−3	1.154101⋅10^−1	2468.7
25	20	ℎ/4	3.884735⋅10^−3	1.155926⋅10^−1	2362.0
25	PODG	N/A	3.884770⋅10^−3	1.156054⋅10^−1	2184.7