*The Finite Element Method Note*

The **strong form** of the boundary-value problem, \((S)\), is stated as follows: \[\begin{aligned}
(S) \left\{
\begin{array}{rl}
Given \ \mathcal{f} : \overline{\Omega} \rightarrow \mathbb{R} \ and \ constants \ q \ and \ h, \ find \ u: \overline{\Omega} \rightarrow \mathbb{R}, \ such \ that \\
u_{,xx}+\mathcal{f}&=0 \ \ \ on \ \Omega\\
u(1)&=q \\
-u_{,x}(0)&=h
\end{array}
\right.\end{aligned}\]

Solution of \((S): \ u(x)=q+(1-x)h+\int_x^1 \left\{ \int_0^y f(z)dz \right\} dy\)

The **weak or variational form** of the boundary-value problem

**Defination**: \(H^1-functions\): functions that satisfy \(\int_0^1 (u_{,x})^2 dx < \infty\)

**Defination**: trial solutions, \(\delta = \{u|u\in H^1, u(1) = q\}\)

**Defination**: weighting functions, \(\nu = \{w|w\in H^1, w(1) = 0\}\)