Minimal finite element spaces for 2𝑚-th-order partial differential equations in 𝑅ⁿ

Source abstract

This paper is devoted to a canonical construction of a family of piecewise polynomials with the minimal degree capable of providing a consistent approximation of Sobolev spaces H m H^m in R n R^n (with n ≥ m ≥ 1 n\ge m\ge 1 ) and also a convergent (nonconforming) finite element space for 2 m 2m -th-order elliptic boundary value problems in R n R^n . For this class of finite element spaces, the geometric shape is n n -simplex, the shape function space consists of all polynomials with a degree not greater than m m , and the degrees of freedom are given in terms of the integral averages of the normal derivatives of order m − k m-k on all subsimplexes with the dimension n − k n-k for 1 ≤ k ≤ m 1\le k\le m . This sequence of spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases. The finite element spaces constructed in this paper constitute the only class of finite element spaces, whether conforming or nonconforming, that are known and proven to be convergent for the approximation of any 2 m 2m -th-order elliptic problems in any R n R^n , such that n ≥ m ≥ 1 n\ge m\ge 1 . Finite element spaces in this class recover the nonconforming linear elements for Poisson equations ( m = 1 m=1 ) and the well-known Morley element for biharmonic equations ( m = 2 m=2 ).