It is proved mathematically that the cell volume is kept to be proportional to the monitor function when the grids are adapted to the solution at each time step. In particular, the moving grid has positive Jacobian and will not fold into itself. The moving grid method is then applied to a driven cavity flow. The results are compared to a benchmark for this problem. It is shown that the grid indeed follows the monitor function closely and that the method appears to have achieved the same accuracy with far less nodes.
Further research is required on constructing the monitor functions to make the method practical.