Transform gradient to reference element

Question

Minimal example of the problem

How can you transform the gradient to the reference element?

Answer 1

General

The reference and the global basis functions are related by

\begin{equation} \hat \varphi_i (\hat x) = \varphi_{n_i} (F_K (\hat x)) = \varphi_{n_i} (B_K \hat{x} + b_K), i = 1, 2, 3. \end{equation}

This yields naturally relation between the gradient too. Assume gradients are column vectors too. Then, it holds too

\begin{equation} \hat \nabla \hat \varphi_i (\hat x) = \hat \nabla \varphi_{n_i} (F_K (\hat x)) \\ = \nabla \varphi_{n_i} ( F_K (\hat x) ) \cdot \hat \nabla F_K (\hat x) \\ = \nabla \varphi_{n_i} ( F_K (\hat x) ) \cdot \hat \nabla (B_K \hat x + b_K) \\ = \nabla \varphi_{n_i} (x) \cdot B_K \\ = B_K^T \nabla \varphi_{n_i} (x) \end{equation} and so \begin{equation} \nabla \varphi_{n_i} (x) = B_K^{-T} \hat \nabla \hat \varphi_i (\hat x). \end{equation}

Consider $K_1$

\begin{equation} \varphi_1 (\hat x) = \varphi_2 (B_K \hat x + b_K) \end{equation}

where

\begin{equation} B_K = [n_2 - n_1; n_3 - n_1] \end{equation}

and take

\begin{equation} b_K = n_2. \end{equation}

I think we just provided ways how to transform the gradient to the reference element.

Sources

1. My lecture notes of Course Element methods, page 7 and Equation 13.