  We present a novel approach for high-order accurate numerical differentiation
on unstructured meshes of quadrilateral elements. To differentiate a given
function, an auxiliary function with greater smoothness properties is defined
which when differentiated provides the derivatives of the original function.
The method generalises traditional finite difference methods to meshes of
arbitrary topology in any number of dimensions for any order of derivative and
accuracy. We demonstrate the accuracy of the numerical scheme using dual
quadrilateral meshes and a refinement method based on subdivision surfaces. The
scheme is applied to the solution of a range of partial differential equations,
including both linear and nonlinear, and second and fourth order equations.
