$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\uexd}[1]{{u_{\small\mbox{e}, #1}}}
\newcommand{\vex}{{v_{\small\mbox{e}}}}
\newcommand{\vexd}[1]{{v_{\small\mbox{e}, #1}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
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\newcommand{\tp}{\thinspace .}
\newcommand{\Ddt}[1]{\frac{D #1}{dt}}
\newcommand{\E}[1]{\hbox{E}\lbrack #1 \rbrack}
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\newcommand{\xpoint}{\boldsymbol{x}}
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\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
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\newcommand{\V}{\boldsymbol{V}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\F}{\boldsymbol{F}}
\newcommand{\stress}{\boldsymbol{\sigma}}
\newcommand{\strain}{\boldsymbol{\varepsilon}}
\newcommand{\stressc}{{\sigma}}
\newcommand{\strainc}{{\varepsilon}}
\newcommand{\I}{\boldsymbol{I}}
\newcommand{\T}{\boldsymbol{T}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
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\newcommand{\iz}{\boldsymbol{i}_z}
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\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
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\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
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$$
Example: linear approximation; solution
$$
\begin{align}
A_{0,0} &= (\baspsi_0,\baspsi_0) = \int_1^21\cdot 1\, dx = 1\\
A_{0,1} &= (\baspsi_0,\baspsi_1) = \int_1^2 1\cdot x\, dx = 3/2\\
A_{1,0} &= A_{0,1} = 3/2\\
A_{1,1} &= (\baspsi_1,\baspsi_1) = \int_1^2 x\cdot x\,dx = 7/3
\end{align}
$$
$$
\begin{align}
b_1 &= (f,\baspsi_0) = \int_1^2 (10(x-1)^2 - 1)\cdot 1 \, dx = 7/3\\
b_2 &= (f,\baspsi_1) = \int_1^2 (10(x-1)^2 - 1)\cdot x\, dx = 13/3
\end{align}
$$
Solution of 2x2 linear system:
$$
\begin{equation}
c_0 = -38/3,\quad c_1 = 10,\quad u(x) = 10x - \frac{38}{3}
\end{equation}
$$