Study guide: Introduction to finite element methods

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

Comparison of the methods

Exact solution: $ u(x)=x(L-x) $

Galerkin or least squares ($ N=0 $): $ u(x)=8L^2\pi^{-3}\sin (\pi x/L) $

Collocation method ($ N=0 $): $ u(x)=2L^2\pi^{-2}\sin (\pi x/L) $.

>>> import sympy as sp
>>> # Computing with Dirichlet conditions: -u''=2 and sines
>>> x, L = sp.symbols('x L')
>>> e_Galerkin = x*(L-x) - 8*L**2*sp.pi**(-3)*sp.sin(sp.pi*x/L)
>>> e_colloc = x*(L-x) - 2*L**2*sp.pi**(-2)*sp.sin(sp.pi*x/L)

>>> # Verify max error for x=L/2
>>> dedx_Galerkin = sp.diff(e_Galerkin, x)
>>> dedx_Galerkin.subs(x, L/2)
0
>>> dedx_colloc = sp.diff(e_colloc, x)
>>> dedx_colloc.subs(x, L/2)
0

# Compute max error: x=L/2, evaluate numerical, and simplify
>>> sp.simplify(e_Galerkin.subs(x, L/2).evalf(n=3))
-0.00812*L**2
>>> sp.simplify(e_colloc.subs(x, L/2).evalf(n=3))
0.0473*L**2