Higher-order methods can accurately approximate PDEs with less computational cost if the PDE solution, u, is sufficiently smooth.  Unfortunately, many (if not most) applications of PDE models involved solutions which are not regular.  Independent of the particular discretization method (e.g. finite element, finite difference, finite volume), the error in the discrete solution, u_h, will converge as the mesh is refined in the following manner,

|| u_h-u ||_X   =  O(h^r)      as    h→ 0

where h is a characteristic length of the elements in the mesh, r is the rate of convergence of the solution error as measured in the X norm.  For finite element methods using polynomials with order k (e.g. k=1 are linear polynomials in an element) and for norms involving only the solution error (e.g. L^p, so called "Lebesgue spaces"), then if the solution is sufficient smooth, a typical theoretical result gives,

|| u_h-u ||_L^p   =  O(h^k+1)      as    h→ 0

However, the sufficiently smooth criteria is critical, and for higher polynomial order the solution needs to have more smoothness in order to realize this optimal result.  Essentially, without the use of specialized discretization techniques, the solution u must have at least k continuous derivatives in order to satisfy the smoothness requirement.   One of the more general approaches which can lessen the impact solution irregularity is adapting the mesh; however, without mesh adaptation, the faster rates of convergence possible with higher-order methods will be limited by the solution smoothness.

Summarizing:

• Higher-order discretizations require increasing smoothness of the analytic solution in order for the discretization to achieve the intended higher-order convergence
• This is true for essentially all discretization methods, in particular finite difference, finite volume, and finite element methods
• One method to decrease the impact of solution irregularity is mesh adaptation