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 φi \basphi_i and insert the Dirichlet condition in the linear system

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