hp-FEM is a generalization of the finite element method (FEM) for solving partial differential equations numerically based on piecewise-polynomial approximations.
hp-FEM originates from the discovery by Barna A. Szabó and Ivo Babuška that the finite element method converges exponentially fast when the mesh is refined using a suitable combination of h-refinements (dividing elements into smaller ones) and p-refinements (increasing their polynomial degree).
This is illustrated in the figure below, where a one-dimensional Poisson equation with zero Dirichlet boundary conditions is solved on two different meshes.
On the contrary, small low-order elements can capture small-scale features such as singularities much better than large high-order ones.
[14] As soon as it is harder to program and parallelize hp-FEM compared to h-FEM, the convergence excellence of hp-refinement may become impractical.
Analogously, splitting a hexahedron into eight sub-elements and varying their polynomial degrees by at most two yields 3^8 = 6,561 refinement candidates.
Standard FEM error estimates providing one constant number per element is not enough to guide automatic hp-adaptivity.