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

Table 3

Comparison of the error between the ROM-calculated velocity 𝐮^𝑟 and the full Galerkin approximation 𝐮^ℎ in Experiment 3 for the POD-Galerkin approximation and various choices of 𝑅,𝛼 in the POD-VMS model.

𝑟	𝑅	𝛼	‖𝐮^ℎ−𝐮^𝑟‖0,2	‖𝐮^ℎ−𝐮^𝑟‖1,2	Time (s)
20	5	ℎ	6.879882⋅10^−2	2.291823	1366.0
20	10	ℎ	7.077484⋅10^−2	2.364495	1186.8
20	15	ℎ	7.366113⋅10^−2	2.629697	1071.5
20	5	4ℎ	6.855368⋅10^−2	2.281231	1349.9
20	10	4ℎ	7.092864⋅10^−2	2.355982	1206.5
20	15	4ℎ	7.368741⋅10^−2	2.629806	1093.8
20	5	ℎ/4	6.991190⋅10^−2	2.365337	1358.6
20	10	ℎ/4	7.110597⋅10^−2	2.423825	1190.6
20	15	ℎ/4	7.363914⋅10^−2	2.631096	1089.8
20	PODG	N/A	7.331827⋅10^−2	2.645780	985.5
25	5	ℎ	6.883794⋅10^−2	2.293204	2065.5
25	10	ℎ	7.082168⋅10^−2	2.365916	1805.9
25	15	ℎ	7.368019⋅10^−2	2.630690	1620.3
25	20	ℎ	7.371688⋅10^−2	2.638875	1508.2
25	5	4ℎ	6.856039⋅10^−2	2.281288	2087.6
25	10	4ℎ	7.093844⋅10^−2	2.356012	1819.7
25	15	4ℎ	7.368747⋅10^−2	2.629795	1634.1
25	20	4ℎ	7.369346⋅10^−2	2.637818	1521.5
25	5	ℎ/4	7.009100⋅10^−2	2.375302	2084.2
25	10	ℎ/4	7.129551⋅10^−2	2.433451	1821.5
25	15	ℎ/4	7.378634⋅10^−2	2.639321	1635.0
25	20	ℎ/4	7.384709⋅10^−2	2.645075	1525.0
25	PODG	N/A	7.432488⋅10^−2	2.676415	1411.8
30	5	ℎ	6.881790⋅10^−2	2.316679	2947.2
30	10	ℎ	7.069041⋅10^−2	2.388068	2605.5
30	15	ℎ	7.333944⋅10^−2	2.645757	2337.7
30	20	ℎ	7.364633⋅10^−2	2.657557	2154.8
30	25	ℎ	7.415144⋅10^−2	2.696152	2043.3
30	5	4ℎ	6.856142⋅10^−2	2.304418	2963.9
30	10	4ℎ	7.080407⋅10^−2	2.378906	2624.2
30	15	4ℎ	7.332959⋅10^−2	2.643993	2366.3
30	20	4ℎ	7.342115⋅10^−2	2.655854	2177.5
30	25	4ℎ	7.414186⋅10^−2	2.695809	2062.6
30	5	ℎ/4	7.004272⋅10^−2	2.400298	2962.4
30	10	ℎ/4	7.117844⋅10^−2	2.456943	2627.9
30	15	ℎ/4	7.352618⋅10^−2	2.658138	2362.4
30	20	ℎ/4	7.365253⋅10^−2	2.667431	2177.1
30	25	ℎ/4	7.418287⋅10^−2	2.698117	2063.2
30	PODG	N/A	7.427912⋅10^−2	2.705275	1950.9