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

Table 1

Comparison of the error between the ROM-calculated velocity 𝐮^𝑟 and the true solution 𝐮 in Experiment 1 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	6.981238⋅10^−3	2.422689⋅10^−1	811.1
10	8	ℎ	6.985801⋅10^−3	2.615740⋅10^−1	765.6
10	5	4ℎ	6.978340⋅10^−3	2.388691⋅10^−1	795.7
10	8	4ℎ	6.984956⋅10^−3	2.614661⋅10^−1	761.9
10	5	ℎ/4	6.987342⋅10^−3	2.524964⋅10^−1	870.6
10	8	ℎ/4	6.989157⋅10^−3	2.612289⋅10^−1	836.5
10	PODG	N/A	6.992952⋅10^−3	2.597919⋅10^−1	698.9
15	5	ℎ	6.978064⋅10^−3	2.422777⋅10^−1	1293.6
15	10	ℎ	6.994673⋅10^−3	2.594528⋅10^−1	1140.7
15	5	4ℎ	6.975261⋅10^−3	2.387484⋅10^−1	1274.7
15	10	4ℎ	6.992465⋅10^−3	2.594345⋅10^−1	1114.8
15	5	ℎ/4	6.987577⋅10^−3	2.527826⋅10^−1	1313.9
15	10	ℎ/4	7.003224⋅10^−3	2.600876⋅10^−1	1161.4
15	PODG	N/A	7.017659⋅10^−3	2.621746⋅10^−1	992.6
20	5	ℎ	6.979928⋅10^−3	2.424063⋅10^−1	1999.8
20	10	ℎ	6.994185⋅10^−3	2.593764⋅10^−1	1736.9
20	15	ℎ	7.003329⋅10^−3	2.610204⋅10^−1	1588.1
20	5	4ℎ	6.9757314⋅10^−3	2.387778⋅10^−1	2014.3
20	10	4ℎ	6.992233⋅10^−3	2.593844⋅10^−1	1757.9
20	15	4ℎ	7.007108⋅10^−3	2.613092⋅10^−1	1603.9
20	5	ℎ/4	6.986873⋅10^−3	2.526078⋅10^−1	2017.9
20	10	ℎ/4	6.998495⋅10^−3	2.597420⋅10^−1	1762.3
20	15	ℎ/4	7.002788⋅10^−3	2.609968⋅10^−1	1609.2
20	PODG	N/A	7.003565⋅10^−3	2.611864⋅10^−1	1419.3
25	5	ℎ	6.982112⋅10^−3	2.426835⋅10^−1	2946.9
25	10	ℎ	6.993659⋅10^−3	2.594056⋅10^−1	2683.4
25	15	ℎ	7.003330⋅10^−3	2.609999⋅10^−1	2429.5
25	20	ℎ	7.002530⋅10^−3	2.611219⋅10^−1	2232.8
25	5	4ℎ	6.973241⋅10^−3	2.387802⋅10^−1	2938.8
25	10	4ℎ	6.989292⋅10^−3	2.593093⋅10^−1	2588.1
25	15	4ℎ	7.006223⋅10^−3	2.612313⋅10^−1	2354.9
25	20	4ℎ	7.002944⋅10^−3	2.611499⋅10^−1	2192.6
25	5	ℎ/4	6.989893⋅10^−3	2.528071⋅10^−1	3147.3
25	10	ℎ/4	6.999151⋅10^−3	2.597719⋅10^−1	2762.9
25	15	ℎ/4	7.003222⋅10^−3	2.609883⋅10^−1	2505.8
25	20	ℎ/4	7.002503⋅10^−3	2.611149⋅10^−1	2343.9
25	PODG	N/A	7.002484⋅10^−3	2.611319⋅10^−1	2033.2