On one problem of 2D regular grid generation based on mappings

To study the known problem of regular grid generation by the Winslow method in a rectangular domain with a boundary kink known as the backstep, a high-accuracy numerical method for the inverse harmonic mapping of the unit square onto the domain with a certain mapping of the domain boundaries is developed. Behavior of the level line of the mapping which enter the point of the boundary kink is studied. Near the kink, the angle between the boundary and the straight line connecting a point on the level line with the point of the boundary kink is found as the function of the coordinate of the point on the level line in the unit square. It is shown that the level line of the mapping is in tangent to the boundary at the kink point. Near the kink point the mapping is non-quasiisometric. The regular grid of the intersection points between the level lines connected by straight lines contains a self-intersecting cell that remains when the grid step along the boundary decreases. Basing on the universal elliptical equations reproducing any nondegenerate mapping of the parametric rectangle onto a given domain, it is suggested a simple two-parametric control of grid nodes in the backstep that allows one to control effectively the slope angle of the grid line entering the point of the kink, thereby removing escape of grid lines from the domain boundary. In the case of the small number of grid points $31\times 31$, the nondegenerate grid is generated by selection of a suitable value of one parameter. When increasing the number of grid points 8 times in both direction (the grid $241\times 241$), the non-convex cells appear within the domain which are easily removed by using the variational barrier method. Another possibility to avoid non-convex cells is to decrease the grid dimension along the second direction (the grid $241\times 121$).