Suppose for a grid-based calculation a grid is used such that the grid Jacobian is discontinuous. For example, in 1D, for a domain $x \in$ [0,1], one half of the domain is covered uniformly by twice as many grid points as the other side of the domain (and this always remains so as the number of grid points is increased). This grid can be illustrated by this plot, showing the x coordinate of the grid point vs. its index normalized to the total number of grid points.
The Jacobian for this grid, representing the transformation of the x coordinate to the grid index is clearly discontinuous at x=1/2. What would be the implications of using such grids for finite-difference of finite-volume calculations such as solving ODEs or PDEs? Would it reduce the grid convergence to the first order? Or not necessarily? Can it lead to losing grid convergence altogether? Would a finite-element (or spectral element) method have an advantage on discontinuous grids, compared to finite difference or finite volume based methods?