## MIT Benchmark

#### Time step and mesh information

The time step is chosen so that there are enough data points in one oscillation of the resulting variables to graphically postprocess all quantities and so that smaller time steps do not significantly improve the solutions with respect to the quantitative measurements. After comparison with the results from Davis [3], Gresho [4], Turek [5] and Le Quéré [6] we choose approximately 34 time steps in one oscillation which corresponds to ΔT=0.1 as time step size. But before, we need to simulate the problem for a very long time (up to time t=1500) until we find a steady oscillation. To do this, we simulate by using a larger time step (ΔT=0.5) and a coarser mesh (2 times refinement from the above coarse mesh). The steady oscillation can be seen from the figure below. Then, starting from the last solution at time t=1500, computation for different meshes with a smaller time step (ΔT=0.1) is undergo for another 100 time unit.

Temperature oscillations at point 1

The plotting data of the above figure can be found here.

Several meshes have been used to perform the spatial discretization. The coarse mesh has approximately 1:5 x-to-y ratio of grid points and decreases gradually towards the walls. We apply local refinement on elements attached to both walls after several regular refinements. Note that regular refinement doubles the number of the total elements in both x and y-direction. The figure below describes how the local refinement is generated for some exemplary meshes. The meshes are denoted by '$n$R_a$i$' for $i$ local refinement steps after $n$ regular refinements.

Several hierarchies and types of meshes

Our very refined mesh, 4R_a3, has total number of elements which are comparable with Davis [3], approximately half from Gresho [4], and one-third from Turek [5] as can be seen from table below.

 Mesh Elements Nodes Edges Dof AuthorMesh Turek128 x 704 Davis83 x 403 Gresho105 x 481 Le Quéré48 x 180 2R 1408 1513 2920 21747 2R_a1 1936 2043 3978 29679 2R_a5 17776 17891 35666 267327 3R 5632 5841 11472 85731 3R_a1 6688 6899 13586 101583 3R_a4 21472 21689 43160 323379 4R 22528 22945 45472 340419 4R_a1 24640 25059 49698 372111 4R_a3 37312 37735 75046 562215

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