# Finite element method for the construction of the approximation of the solution

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!! Given the following two-point problem:
$$-y''(x)+(by)'(x)=f(x), \forall x \in [0,1]$$
$$y(0)=0, y'(1)=my(1)$$
where $b \in C^1([0,1];R), f \in C([0,1];R)$ and $m \in R$ a constant.
Give a finite element method for the construction of the approximation of the solution $y$ of the problem above, where the finite element space ($S$) consists of continuous and partially linear functions.

My idea is the following:
$u \in S:$
$$-\int_0^1{u''g}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$$-u'g|_0^1+ \int_0^1{u'g'}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$$-mu(1)+g(1)+ \int_0^1{u'g'}dx+ \int_0^1{bu'g}dx= \int_0^1{fg}dx$$
$\forall g \in S$

Could you tell me if this is correct?

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#### mathmari

##### Well-known member
MHB Site Helper
To find the method we take a function $g$ of $S$, right? Does this function satisfy the conditions of the problem? I mean $g(0)=0, g'(1)=mg(1)$... Or is there an other way to find the method?