Study guide: Introduction to finite element methods

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Method 2: Use all $ \basphi_i $ and insert the Dirichlet condition in the linear system

Now $ \baspsi_i=\basphi_i $, $ i=0,\ldots,N=N_n $

$ \basphi_N(L)\neq 0 $, so $ u'(L)\basphi_N(L)\neq 0 $

However, the term $ u'(L)\basphi_N(L) $ in $ b_N $ will be erased when we insert the Dirichlet value in $ b_N=D $

We can therefore forget about the term $ u'(L)\basphi_i(L) $!

Boundary terms $ u'\basphi_i $ at points $ \xno{i} $ where Dirichlet values apply can always be forgotten.

$$ \begin{equation*} u(x) = \sum_{j=0}^{N=N_n} c_j\basphi_j(x) \end{equation*} $$ $$ \begin{equation*} \sum_{j=0}^{N=N_n}\left( \int_0^L \basphi_i'(x)\basphi_j'(x) dx \right)c_j = \int_0^L f(x)\basphi_i(x) dx - C\basphi_i(0) \end{equation*} $$

Assemble entries for $ i=0,\ldots,N=N_n $ and then modify the last equation to $ c_N=D $

Method 2: Use all \( \basphi_i \) and insert the Dirichlet condition in the linear system