Study guide: Introduction to finite element methods

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$\newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} \newcommand{\x}{\boldsymbol{x}} \newcommand{\X}{\boldsymbol{X}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\e}{\boldsymbol{e}} \newcommand{\f}{\boldsymbol{f}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\Ix}{\mathcal{I}_x} \newcommand{\Iy}{\mathcal{I}_y} \newcommand{\Iz}{\mathcal{I}_z} \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifd}{{I_d}} % for FEM \newcommand{\Ifb}{{I_b}} % for FEM \newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}} \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\refphi}{\tilde\basphi} \newcommand{\psib}{\boldsymbol{\psi}} \newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)} \newcommand{\xno}[1]{x_{#1}} \newcommand{\Xno}[1]{X_{(#1)}} \newcommand{\xdno}[1]{\boldsymbol{x}_{#1}} \newcommand{\dX}{\, \mathrm{d}X} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s}$

With a $B(x)$ , $u\not\in V$ , but $\sum_{j}c_j\baspsi_j\in V$

$\sequencei{\baspsi}$ is a basis for $V$

$\sum_{j\in\If}c_j\baspsi_j(x)\in V$

But $u\not\in V$ !

Reason: say $u(0)=C$ and $u\in V$ ; any $v\in V$ has $v(0)=C$ , then $2u\not\in V$ because $2u(0)=2C$ (wrong value)

When $u(x) = B(x) + \sum_{j\in\If}c_j\baspsi_j(x)$ , $B\not\in V$ (in general) and $u\not\in V$ , but $(u-B)\in V$ since $\sum_{j}c_j\baspsi_j\in V$

With a B(x) B(x) , u∉V u\not\in V , but ∑jcjψj∈V \sum_{j}c_j\baspsi_j\in V

With a $B(x)$ , $u\not\in V$ , but $\sum_{j}c_j\baspsi_j\in V$