2.11 The Finite Element Method for Two-Dimensional Diffusion

2.11.1 Overview

In this lecture, we will consider the finite element approximation of the two-dimensional diffusion problem,

\[\nabla \cdot \left(k \nabla T\right) + f = 0. \label{equ:dif2d}\] (2.259)

As in the previous discussion of the method of weighted residuals and the finite element method, the approximate solution will have the form

\[\tilde{T}(x,y) = \sum _{j=1}^{N} a_ j \phi _ j(x,y),\] (2.260)

where \(\phi _ j(x,y)\) are the known basis functions and the \(a_ j\) are the unknown coefficients to be determined for the specific problem. Following the Galerkin method of weighted residuals, we will weight Equation (2.259) by one of the basis functions and integrate the diffusion term by parts to give the following weighted residual:

\[R_ i \equiv \int _{\delta \Omega } \phi _ i\, k\nabla \tilde{T}\cdot \vec{n}\, ds - \int _{\Omega } \nabla \phi _ i \cdot \left(k\nabla \tilde{T}\right)\, dA + \int _{\Omega } \phi _ i f\, dA = 0. \label{equ:mwr_ dif2d}\] (2.261)