hp-FEM

Remarks

The $h$-version is the case of constant $p$ and finer meshes, and the $p$-version is the case of a fixed mesh and growing degrees. Both are special cases of the finite element method in this setting.

Example: on $\Omega = (0, 1)$ with a solution behaving like $x^\lambda$ near $0$, with $0 < \lambda < 1$, take a geometric mesh with nodes $0, \sigma^{n-1}, \ldots, \sigma, 1$ for some $0 < \sigma < 1$. Take degrees that grow linearly away from $0$. For such geometric meshes, the energy error decays exponentially in a root of the number of unknowns $N$. In two dimensions the rate is $\exp(-b N^{1/3})$ for piecewise analytic data on polygons.

Non-example: let $V = H^1_0(\Omega)$ and take all functions that are polynomials of degree at most $p_K$ on each $K$, with no continuity across elements. A function with a jump across an interior face is not in $H^1(\Omega)$, so this set fails the condition $v \in V$. Such spaces are used by hp discontinuous Galerkin methods instead.