The finite difference method doesn’t postulate explicitly any specific shape of the unknown field. As we are concerned with partial differential equations, exact derivatives are replaced by an approximation based on neighbouring values of the unknown (still denoted as p):
P,+1 — Pi-1
W/, 2L
where the subscript i denotes the cell number and L denotes the cell size. For an orthogonal mesh, such derivatives are easily generalised to variable cell dimensions. However, non-orthogonal meshes pose problems that are highly difficult to solve and are generally not used. Boundary conditions have then to be modelled by the juxtaposition of orthogonal cells, giving a kind of stepped edge for oblique or curved boundaries. Similarly, local refinement of the mesh induces irreducible global refinement. These aspects are the most prominent drawbacks of the finite difference method compared to the finite element one. On the other hand computing time is generally much lower with finite differences then with finite elements.
