# A general approach to transforming finite elements

## TL;DR

This paper introduces a general method for transforming finite element bases, enabling the use of reference elements for a broader class of finite elements that previously could not be handled in this way.

## Contribution

It develops a unified approach to map bases for complex finite elements like Hermite and Argyris, expanding the applicability of reference element techniques.

## Key findings

- Provides a new framework for finite element basis transformation
- Enables efficient implementation of complex finite elements
- Broadens the class of finite elements compatible with reference element methods

## 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 generalizated approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward.