A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

A unified construction of canonical H m H^m -nonconforming finite elements is developed for n n -dimensional simplices for any m , n ≥ 1 m, n ≥ 1 . Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25–43] is maintained when m ≤ n m ≤ n . In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on m m . The corresponding nonconforming finite element method applies to 2 m 2m -th order elliptic problems, with numerical results for m = 3 m=3 and m = 4 m=4 in 2D supporting the theoretical analysis.