Finite Element Method - Discretization

Discretization

P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite dimensional linear problem:

Find such that

with a finite dimensional version:

(3) Find such that

where is a finite dimensional subspace of . There are many possible choices for (one possibility leads to the spectral method). However, for the finite element method we take to be a space of piecewise polynomial functions.