Finite element differential forms on curvilinear cubic meshes and their approximation properties

Abstract

We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L 2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence. Mathematics Subject Classification (2010)

Finite element differential forms on cubical meshes

“…An analogue to the P − r Λ k complex of elements for cubical meshes may be easily constructed via a tensor product construction. For an explicit description in n dimensions and for all form degrees k, see [3]. This includes the tensor product Lagrange, or Q r , elements for 0-forms, the rectangular RT elements for 1-forms in 2-D, and the 3-D generalizations of them given in [9].…”

Stokes elements on cubic meshes yielding divergence-free approximations

“…Proof The proof is by induction on m. The case m = 0 is clearly true since (1) and S (2) . Denote by ∇ S and ∇ S (i) the surface gradient of S and S (i) , respectively.…”

Automated Generation and Symbolic Manipulation of Tensor Product Finite Elements

“…Product complexes using differential forms. This section summarizes Arnold, Boffi, and Bonizzoni [6] by restating the results of subsection 2.4 and subsection 2.5 in the language of differential forms, which can be considered a generalization of scalar and vector fields.…”

“…Arnold, Boffi, and Bonizzoni [6] generalize finite element exterior calculus to cells which can be expressed as geometric products of simplices. They also describe a specific complex of finite element spaces on hexahedra (and, implicitly, quadrilaterals).…”

“…This covers many, though not all, of the common finite element spaces on product cells. We pay special attention to element families relevant to finite element exterior calculus, a mathematical framework that leads to stable mixed finite element discretizations of PDEs [7,8,6]. This paper therefore describes some of the extensions to the Firedrake code-generation pipeline to enable the solution of finite element problems on cells which are products of simplices.…”