A general approach to transforming finite elements

Abstract

The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way. This paper gives a generalized approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward. This approach, developed through a sequence of examples of increasing complexity, requires one to study the relationship between the function space and degrees of freedom, or nodes, on a generic cell and the transformation of the corresponding entities on a reference cell. When the function space is preserved under mapping, one must be able to express the pushed-forward finite element nodes as linear combinations of the reference element finite element nodes. The transpose of this linear transformation maps the pull-back of the reference element basis functions to the desired finite element basis functions. Generically, developing this transformation for elements such as Morley and Hermite involves completing the set of finite element nodes, although the process is simplified in concrete ways when the finite elements form affine- or affine-interpolation equivalent families such as Lagrange or Hermite. When the finite element function space is not preserved under the pull-back, such as in the case of the Bell element, one applies the theory to an enriched finite element with a larger (but preserved) function space and additional nodes.