3.1 Local approximations

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3.1 Local approximations

In the finite element method, the solution of a PDE is approximated by low-order polynomials on local elements. The local elements constitute the mesh; typical elements used are triangles and quadrilaterals in 2D, and tetrahedra and hexahedra in 3D.

To give a simple example, consider a triangular mesh in 2D (see figure). We concentrate on the single triangle with corner nodes $\{i,j,k\}$ , and let the values of

at the nodes be $\{u_i,u_j,u_k\}$ . We approximate

within the local element by

$\begin{displaymath} \bar{u}=\left[ \begin{array} {ccc} \phi_i(x,y) & \phi_j(x,y)... ...ight]\left[ \begin{array} {l} u_i\\ u_j\\ u_k\end{array}\right]\end{displaymath}$

where $\{\phi_i, \phi_j, \phi_k\}$ are interpolation functions. In the simplest case, these are linear polynomials such that

$\begin{displaymath} \begin{array} {llll} \phi_i=1, & \phi_j=0, & \phi_k=0, & \mb... ..._i=0, & \phi_j=0, & \phi_k=1, & \mbox{ at node }k\\ \end{array}\end{displaymath}$

For example, if the local element is the triangle with nodes at

and

, the three linear interpolation functions are

$\begin{displaymath} \begin{array} {l} \phi_1=1-x\\ \phi_2=x-y\\ \phi_3=y\\ \end{array}\end{displaymath}$

and, given nodal values

, the linear approximation to

in the element is

$\begin{displaymath} u(x,y)=\left[ \begin{array} {ccc} 1-x & x-y & y\end{array}\right]\left[ \begin{array} {l} u_i\\ u_j\\ u_k\end{array}\right]\end{displaymath}$