A Construction of $C^r$ Conforming Finite Element Spaces in Any Dimension

Abstract

This paper proposes a construction of local C r interpolation spaces and C r conforming finite element spaces with arbitrary r in any dimension. It is shown that if k ≥ 2 d r +1 the space P k of polynomials of degree ≤ k can be taken as the shape function space of C r finite element spaces in d dimensions. This is the first work on constructing such C r conforming finite elements in any dimension in a unified way. It solves a long-standing open problem in finite element methods.

Geometric decompositions of the simplicial lattice and smooth finite elements in arbitrary dimension

“…In a recent work [9], Hu, Lin and Wu have solved a long-standing open problem in finite element methods: construction of C m -conforming finite elements on simplexes in arbitrary dimension. It unifies the scattered results [3,18,1] in two dimensions, [19,25] in three dimensions, and [26] in four dimensions.…”

“…It unifies the scattered results [3,18,1] in two dimensions, [19,25] in three dimensions, and [26] in four dimensions. In this paper, we provide a geometric decomposition of the finite element spaces constructed in [9] and consequently give a simplified construction different from [9].…”

“…The continuity of ∂ r n u on faces will imply stronger smoothness on edges and vertices, which is known as the supersmoothness [17,7]. Indeed in [9], the authors constructed C m -conforming finite elements under the following requirement on the C r smoothness at -dimensional sub-simplexes and the degree of polynomial is exponential in n:…”

“…, α n ) ∈ N 0:n | |α| = k as the multi-index set with fixed length k. Due to the one-to-one mapping between the Bernstein polynomial λ α , where λ is the barycentric coordinate, and the lattice node α ∈ T n k , the key in the construction is a non-overlapping decomposition (partition) of the simplicial lattice in which each component will be used to determine the normal derivatives at each lower dimensional sub-simplex. In [9], a purely algebraic and combinatory approach is used to prove the existence of such partition. In this paper, a geometric approach will be proposed.…”

On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

“…The construction of sych approximation spaces with higher regularity is normally deemed a difficult task because it requires a set of basis functions with such global regularity. Examples in this direction can be found all along the history of finite elements from the oldest works in the sixties of the last century, e.g., [12,26,40] to to the most recent attempts in [44,45,63,64]. Despite its intrinsic difficulty, designing approximations with global 𝐶 1 -or higher regularity is still a major research topic.…”