  In this work, we propose a family of interior penalty nonconforming finite
element methods for $2m$-th order partial differential equations in
$\mathbb{R}^n$, for any $m \geq 0$, $n \geq 1$. For the nonconforming finite
element, the shape function space consists of all polynomials with a degree not
greater than $m$ and is hence minimal. This family of finite element spaces has
some natural inclusion properties as in the corresponding Sobolev spaces in the
continuous cases. By applying the interior penalty to the bilinear form, we
establish quasi-optimal error estimates in the energy norm, provided that the
corresponding conforming relatives exist. These theoretical results are further
validated by the numerical tests.
