def fine_mesh_size(mesh):
    """
    Returns the number of cells in the x- and y-directions for the fine mesh 
    """  
    if mesh.grid_size() is None:
        """
        No initial grid
        """
        nx_0, ny_0 = 1,1
        l_max = mesh.depth()
    else:
        nx_0, ny_0 = mesh.grid_size()
        l_max = mesh.depth()-1
    
    nx, ny = nx_0*2**l_max, ny_0*2**l_max
    return nx, ny