How are you all~ Could you tell me the reference regarding C-grid and O-grid mesh ? How to include branch cut into the calculation is most puzzling. Thank you for your advice.

Treat the cut in the grid as a 'periodic' boundary condition ie, exchange the values of the neighbouring cells along the boundary.

I do not know what you mean by C-grid and O-grid, but if I were doing a computation in the complex plane with a branch cut, I would try the following. Assume the cut lies along the x-axis. I would construct the mesh of two blocks, one lying above the x-axis and one below the x-axis. Along the interface without the cut I would set (constrain) (f+)=(f-) at each of the nodes. Along the cut I would set (f+)-(f-)=delta at each node, where (f+) is the value of the complex function at a node above the axis (in block 1, say), (f-) is the value at a node below the axis (in block 2), and delta is the discontinuity across the branch cut (assuming you know this). This could be done after the global matrix is assembled on each block. I don't know what I would do at the branch point.

Chris~ and Jonas~ Thank you so much for your comments. Have a wonderful day~