$$
\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}}
\newcommand{\halfi}{{1/2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Ddt}[1]{\frac{D #1}{dt}}
\newcommand{\E}[1]{\hbox{E}\lbrack #1 \rbrack}
\newcommand{\Var}[1]{\hbox{Var}\lbrack #1 \rbrack}
\newcommand{\Std}[1]{\hbox{Std}\lbrack #1 \rbrack}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\w}{\boldsymbol{w}}
\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
\newcommand{\ii}{\boldsymbol{i}}
\newcommand{\jj}{\boldsymbol{j}}
\newcommand{\kk}{\boldsymbol{k}}
\newcommand{\ir}{\boldsymbol{i}_r}
\newcommand{\ith}{\boldsymbol{i}_{\theta}}
\newcommand{\iz}{\boldsymbol{i}_z}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
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\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
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\newcommand{\Real}{\mathbb{R}}
\newcommand{\Integerp}{\mathbb{N}}
\newcommand{\Integer}{\mathbb{Z}}
$$
The least squares method
- Extend the ideas from the vector case: minimize the (square) norm
of the error.
- What norm? \( (f,g) = \int_\Omega f(x)g(x)\, dx \)
$$
\begin{equation}
E = (e,e) = (f-u,f-u) = (f(x)-\sum_{j\in\If} c_j\baspsi_j(x), f(x)-\sum_{j\in\If} c_j\baspsi_j(x))
\tag{7}
\end{equation}
$$
$$
\begin{equation}
E(c_0,\ldots,c_N) = (f,f) -2\sum_{j\in\If} c_j(f,\baspsi_i)
+ \sum_{p\in\If}\sum_{q\in\If} c_pc_q(\baspsi_p,\baspsi_q)
\end{equation}
$$
$$
\begin{equation*}
\frac{\partial E}{\partial c_i} = 0,\quad i=\in\If
\end{equation*}
$$
After computations identical to the vector case, we get a linear system
$$
\begin{align}
\sum_{j\in\If}^N A_{i,j}c_j &= b_i,\quad i\in\If
\tag{8}\\
A_{i,j} &= (\baspsi_i,\baspsi_j)
\tag{9}\\
b_i &= (f,\baspsi_i)
\tag{10}
\end{align}
$$