2.11 The Finite Element Method for Two-Dimensional Diffusion

2.11.5 Integration in the Reference Element

Measurable Outcome 2.20

The reference element can also be used to evaluate integrals. For example, consider the evaluation of the forcing function integral within an element:

\[\int _{\delta \Omega _ k} w(\vec{x})\, f(\vec{x})\, dA.\] (2.285)

In transforming the integral from \((x,y)\) to \((\xi _1,\xi _2)\), the differential area of integration must be transformed using the following result:

\[dA = dx\, dy = J d\xi _1\, d\xi _2 = J\, dA_{\xi }. \label{equ:Ax_ to_ Axi}\] (2.286)

Thus, the integrals can now be evaluated in reference element space,

\[\int _{\Omega _{\xi }} w(\vec{x}(\vec{\xi }))\, f(\vec{x}(\vec{\xi }))\, J dA_{\xi }.\] (2.287)