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notebook.tex
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notebook.tex
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% Default to the notebook output style
% Inherit from the specified cell style.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
% Nicer default font (+ math font) than Computer Modern for most use cases
\usepackage{mathpazo}
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% We will generate all images so they have a width \maxwidth. This means
% that they will get their normal width if they fit onto the page, but
% are scaled down if they would overflow the margins.
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
\else\Gin@nat@width\fi}
\makeatother
\let\Oldincludegraphics\includegraphics
% Set max figure width to be 80% of text width, for now hardcoded.
\renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}}
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionLabelFormat{nolabel}{}
\captionsetup{labelformat=nolabel}
\usepackage{adjustbox} % Used to constrain images to a maximum size
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
% Colors for the hyperref package
\definecolor{urlcolor}{rgb}{0,.145,.698}
\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
\definecolor{citecolor}{rgb}{.12,.54,.11}
% ANSI colors
\definecolor{ansi-black}{HTML}{3E424D}
\definecolor{ansi-black-intense}{HTML}{282C36}
\definecolor{ansi-red}{HTML}{E75C58}
\definecolor{ansi-red-intense}{HTML}{B22B31}
\definecolor{ansi-green}{HTML}{00A250}
\definecolor{ansi-green-intense}{HTML}{007427}
\definecolor{ansi-yellow}{HTML}{DDB62B}
\definecolor{ansi-yellow-intense}{HTML}{B27D12}
\definecolor{ansi-blue}{HTML}{208FFB}
\definecolor{ansi-blue-intense}{HTML}{0065CA}
\definecolor{ansi-magenta}{HTML}{D160C4}
\definecolor{ansi-magenta-intense}{HTML}{A03196}
\definecolor{ansi-cyan}{HTML}{60C6C8}
\definecolor{ansi-cyan-intense}{HTML}{258F8F}
\definecolor{ansi-white}{HTML}{C5C1B4}
\definecolor{ansi-white-intense}{HTML}{A1A6B2}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Additional commands for more recent versions of Pandoc
\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
\newcommand{\ImportTok}[1]{{#1}}
\newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
\newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
\newcommand{\BuiltInTok}[1]{{#1}}
\newcommand{\ExtensionTok}[1]{{#1}}
\newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
\newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatability definitions
\def\gt{>}
\def\lt{<}
% Document parameters
\title{FiniteElementApproach\_1D}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
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\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
\expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}}
\expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
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\expandafter\def\csname PY@tok@sx\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
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\expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}}
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\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}}
\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit}
\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf}
\expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
\expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}}
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\expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
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\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
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\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
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% for compatibility with earlier versions
\def\PYZat{@}
\def\PYZlb{[}
\def\PYZrb{]}
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% Prevent overflowing lines due to hard-to-break entities
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% Setup hyperref package
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breaklinks=true, % so long urls are correctly broken across lines
colorlinks=true,
urlcolor=urlcolor,
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\begin{document}
\maketitle
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}20}]:} \PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Lagrange\PYZus{}polynomial}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{i}\PY{p}{,} \PY{n}{points}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Compute lagrangian interpolation polynomial }
\PY{l+s+sd}{ over domain x at specific points \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{p} \PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{points}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{k} \PY{o}{!=} \PY{n}{i}\PY{p}{:}
\PY{n}{p} \PY{o}{*}\PY{o}{=} \PY{p}{(}\PY{n}{x} \PY{o}{\PYZhy{}} \PY{n}{points}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{)} \PY{o}{/} \PY{p}{(}\PY{n}{points}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{points}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{n}{p}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Lagrange\PYZus{}polynomial\PYZus{}basis}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{points}\PY{p}{,} \PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Compute complete basis of lagrange polynomials }
\PY{l+s+sd}{ at specified points \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k}{return} \PY{p}{[}\PY{n}{Lagrange\PYZus{}polynomial}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{i}\PY{p}{,} \PY{n}{points}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Chebyshev\PYZus{}nodes}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{b}\PY{p}{,} \PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Computes Chebyshev nodes in a specific range\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{k+kn}{from} \PY{n+nn}{math} \PY{k}{import} \PY{n}{cos}\PY{p}{,} \PY{n}{pi}
\PY{k}{return} \PY{p}{[}\PY{l+m+mf}{0.5}\PY{o}{*}\PY{p}{(}\PY{n}{a}\PY{o}{+}\PY{n}{b}\PY{p}{)} \PY{o}{+} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{p}{(}\PY{n}{b}\PY{o}{\PYZhy{}}\PY{n}{a}\PY{p}{)}\PY{o}{*}\PY{n}{cos}\PY{p}{(}\PY{p}{(}\PY{l+m+mf}{2.}\PY{o}{*}\PY{n}{i}\PY{o}{+}\PY{l+m+mf}{1.}\PY{p}{)}\PY{o}{/}\PY{p}{(}\PY{l+m+mf}{2.}\PY{o}{*}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mf}{1.}\PY{p}{)}\PY{p}{)}\PY{o}{*}\PY{n}{pi}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Uniform\PYZus{}nodes}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{b}\PY{p}{,} \PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}Computes uniformly spaced nodes\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{h} \PY{o}{=} \PY{p}{(}\PY{n}{b}\PY{o}{\PYZhy{}}\PY{n}{a}\PY{p}{)} \PY{o}{/} \PY{n}{N}
\PY{k}{return} \PY{p}{[}\PY{n}{a} \PY{o}{+} \PY{n}{i}\PY{o}{*}\PY{n}{h} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{init\PYZus{}elements}\PY{p}{(}\PY{n}{Omega}\PY{p}{,} \PY{n}{N}\PY{p}{,} \PY{n}{n}\PY{p}{,} \PY{o}{*}\PY{o}{*}\PY{n}{kwargs}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Function initializes a 2D array which defines N+1 global }
\PY{l+s+sd}{ 1D finite elements, with n+1 nodes inside each element}
\PY{l+s+sd}{ in the domain Omega = (x0, x1) \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{node\PYZus{}spacing} \PY{o}{=} \PY{n}{kwargs}\PY{o}{.}\PY{n}{get}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{node\PYZus{}spacing}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{uniform}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{if} \PY{p}{(}\PY{n}{node\PYZus{}spacing} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{uniform}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{:}
\PY{n}{nodes} \PY{o}{=} \PY{n}{Uniform\PYZus{}nodes}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{+} \PY{n}{N}\PY{o}{*}\PY{n}{n}\PY{p}{)}
\PY{k}{elif} \PY{p}{(}\PY{n}{node\PYZus{}spacing} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{chebyshev}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{:}
\PY{n}{nodes} \PY{o}{=} \PY{n}{Chebyshev\PYZus{}nodes}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{+} \PY{n}{N}\PY{o}{*}\PY{n}{n}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Initialize elements}
\PY{n}{elements} \PY{o}{=} \PY{p}{[}\PY{p}{[}\PY{n}{i}\PY{o}{\PYZhy{}}\PY{n}{j}\PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{j}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{n}{j}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{]} \PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{return} \PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{)}
\end{Verbatim}
\section{Finite Element Method in 1D using Least Squares
Method}\label{finite-element-method-in-1d-using-least-squares-method}
\subsection{Element matrix}\label{element-matrix}
\subsubsection{Indice notation}\label{indice-notation}
\begin{itemize}
\tightlist
\item
\(e\): Element indice
\item
\(r\): local node indice
\item
\(i\): global node indice
\item
\(i = q(e,r)\): Mapping of local node number \(r\) in element \(e\) to
the global node number \(i\). Mathematical notation for
\(i = elements[e][r]\)
\end{itemize}
\subsubsection{Element matrix
definition}\label{element-matrix-definition}
\((d+1)\times(d+1)\)-Matrix, that stores the local contributions \[
A_{i,j}^{(e)}
= \int_{\Omega^{(e)}} \phi_i \phi_j \text{d}x
= \int_{\Omega^{(e)}} \phi_{q(e,r)} \phi_{q(e,s)} \text{d}x
\]
Let \(I_d = \{0,...,d\}\) be the valid indices of \(r\) and \(s\). We
then introduce the notation \[
\tilde{A}^{(e)} = \left\{\tilde{A}^{(e)}_{r,s}\right\}, \qquad r,s \in I_d
\] For example, \(d=2\) leads to \[
\tilde{A}^{(e)} =
\begin{bmatrix}
\tilde{A}^{(e)}_{0,0} & \tilde{A}^{(e)}_{0,1} & \tilde{A}^{(e)}_{0,2} \\
\tilde{A}^{(e)}_{1,0} & \tilde{A}^{(e)}_{1,1} & \tilde{A}^{(e)}_{1,2} \\
\tilde{A}^{(e)}_{2,0} & \tilde{A}^{(e)}_{2,1} & \tilde{A}^{(e)}_{2,2} \\
\end{bmatrix}
\]
\subsubsection{Affine mapping}\label{affine-mapping}
Instead of computing the integrals in \(\tilde{A}^{(e)}\) for every
element domain \(\Omega^{(e)} = [x_L, x_R]\), a reference element is
introduced over the domain \([-1,1]\).
A linear affine mapping from \(X \in [-1,1]\) to \(x \in \Omega^{(e)}\)
leads to \[
x
= \frac{1}{2} (x_L + x_R) + \frac{1}{2} (x_R - x_L) X
= x_m + \frac{1}{2} h X
\]
The integration over the reference element is a matter of changing the
integration variable from \(x\) to \(X\). Introducing the basis function
\[
\tilde{\phi}_r(X) = \phi_{q(e,r)}\left(x(X)\right)
\] associated with the local node \(r\) in the reference element. The
element matrix entry for the reference element reads then \[
\tilde{A}^{(e)}_{r,s}
= \int_{\Omega^{(e)}} \phi_{q(e,r)} \phi_{q(e,s)} \text{d}x
= \int_{-1}^{1} \tilde{\phi}_r(X) \tilde{\phi}_s(X) \frac{\text{d}x}{\text{d}X} \text{d}X
= \int_{-1}^{1} \tilde{\phi}_r(X) \tilde{\phi}_s(X) \text{det}(J) \text{d}X
\] Hereby the factor \(\frac{\text{d}x}{\text{d}X}\) is the determinant
of the Jacobian matrix of the mapping between the coordinate systemxs.
For 1D problems, this factor reduces to \(\text{det}(J) = h/2\). The
corresponding entries for the element right hand side of the linear
equation system reads \[
\tilde{b}_{r}^{(e)}
= \int_{\Omega^{(e)}} f(x) \phi_{q(e,r)}(x) \text{d}x
= \int_{-1}^1 f\left(x(X)\right) \tilde{\phi}_r(X) \text{det}(J) \text{d}X
\]
The \(\tilde{\phi}_r(X)\) fuctions are the Lagrangian polynomials
defined through the local nodes of the reference element.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}2}]:} \PY{c+c1}{\PYZsh{} https://github.com/hplgit/INF5620/blob/master/src/fem/fe\PYZus{}approx1D.py}
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{k+kn}{from} \PY{n+nn}{scipy} \PY{k}{import} \PY{n}{integrate}
\PY{c+c1}{\PYZsh{}=======================================================}
\PY{c+c1}{\PYZsh{} Implementation of the Finite Element Method in 1D}
\PY{c+c1}{\PYZsh{} for the approximateion of a function f(x)}
\PY{c+c1}{\PYZsh{}=======================================================}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Lagrange\PYZus{}polynomial}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{i}\PY{p}{,} \PY{n}{points}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Compute lagrangian interpolation polynomial }
\PY{l+s+sd}{ over domain x at specific points \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{p} \PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{points}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{k} \PY{o}{!=} \PY{n}{i}\PY{p}{:}
\PY{n}{p} \PY{o}{*}\PY{o}{=} \PY{p}{(}\PY{n}{x} \PY{o}{\PYZhy{}} \PY{n}{points}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{)} \PY{o}{/} \PY{p}{(}\PY{n}{points}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{points}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{n}{p}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{phi\PYZus{}r}\PY{p}{(}\PY{n}{r}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Defines a Lagrangian basis function for the }
\PY{l+s+sd}{ local node r of an reference element of }
\PY{l+s+sd}{ degree d \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{nodes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mf}{1.}\PY{p}{,} \PY{l+m+mf}{1.}\PY{p}{,} \PY{n}{d}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}
\PY{k}{return} \PY{n}{Lagrange\PYZus{}polynomial}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{r}\PY{p}{,} \PY{n}{nodes}\PY{p}{)}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{basis}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Create all basis functions for a reference }
\PY{l+s+sd}{ element of degree d}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{phi} \PY{o}{=} \PY{p}{[}\PY{n}{phi\PYZus{}r}\PY{p}{(}\PY{n}{r}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)} \PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{d}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{return} \PY{n}{phi}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{element\PYZus{}matrix}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Creates the element matrix for an element over}
\PY{l+s+sd}{ its local domain Omega\PYZus{}e=[x0\PYZus{}e, x1\PYZus{}e] and its }
\PY{l+s+sd}{ basis functions stored in phi }
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{n} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{phi}\PY{p}{)}
\PY{n}{A\PYZus{}e} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{n}{n}\PY{p}{)}\PY{p}{)}
\PY{n}{h} \PY{o}{=} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{detJ} \PY{o}{=} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{h}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{,} \PY{n}{s}\PY{p}{]} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{trapz}\PY{p}{(}\PY{n}{phi}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{o}{*}\PY{n}{phi}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{o}{*}\PY{n}{detJ}\PY{p}{,} \PY{n}{X}\PY{p}{)}
\PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{s}\PY{p}{,} \PY{n}{r}\PY{p}{]} \PY{o}{=} \PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{,} \PY{n}{s}\PY{p}{]}
\PY{k}{return} \PY{n}{A\PYZus{}e}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{element\PYZus{}rhs}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{f}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}\PY{p}{:}
\PY{n}{n} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{phi}\PY{p}{)}
\PY{n}{b\PYZus{}e} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{n}\PY{p}{)}
\PY{n}{h} \PY{o}{=} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{detJ} \PY{o}{=} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{h}
\PY{c+c1}{\PYZsh{} Compute mapping from f(x) to f(x(X))}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n+nb}{len}\PY{p}{(}\PY{n}{X}\PY{p}{)}\PY{p}{)}
\PY{n}{f\PYZus{}map} \PY{o}{=} \PY{n}{f}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{b\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{]} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{trapz}\PY{p}{(}\PY{n}{f\PYZus{}map}\PY{o}{*}\PY{n}{phi}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{o}{*}\PY{n}{detJ}\PY{p}{,} \PY{n}{X}\PY{p}{)}
\PY{k}{return} \PY{n}{b\PYZus{}e}
\PY{c+c1}{\PYZsh{} Test the fem\PYZhy{}functions}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{n}{d} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{Omega\PYZus{}e} \PY{o}{=} \PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{X} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{501}\PY{p}{)}
\PY{n}{f} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{:} \PY{n}{x}\PY{o}{*}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{n}{x}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Analytical solutions }
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+se}{\PYZbs{}n}\PY{l+s+s2}{Analytical soltion to element matrix and element rhs (only for d=1): }\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{h} \PY{o}{=} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{xm} \PY{o}{=} \PY{l+m+mf}{0.5} \PY{o}{*} \PY{p}{(}\PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{h: }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{n}{h}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{A[i][i]: }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{n}{h}\PY{o}{/}\PY{l+m+mi}{3}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{A[i][j]: }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{n}{h}\PY{o}{/}\PY{l+m+mi}{6}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{b[0]: }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{p}{(}\PY{l+m+mf}{1.}\PY{o}{/}\PY{l+m+mf}{24.}\PY{p}{)} \PY{o}{*} \PY{n}{h}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3} \PY{o}{+} \PY{p}{(}\PY{n}{xm}\PY{o}{/}\PY{l+m+mf}{6.}\PY{p}{)}\PY{o}{*}\PY{n}{h}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{\PYZhy{}} \PY{p}{(}\PY{l+m+mf}{1.}\PY{o}{/}\PY{l+m+mf}{12.}\PY{p}{)}\PY{o}{*}\PY{n}{h}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{xm}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{h} \PY{o}{+} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{h}\PY{o}{*}\PY{n}{xm}\PY{p}{)}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{b[1]: }\PY{l+s+s2}{\PYZdq{}} \PY{o}{+} \PY{n+nb}{str}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{p}{(}\PY{l+m+mf}{1.}\PY{o}{/}\PY{l+m+mf}{24.}\PY{p}{)} \PY{o}{*} \PY{n}{h}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{3} \PY{o}{\PYZhy{}} \PY{p}{(}\PY{n}{xm}\PY{o}{/}\PY{l+m+mf}{6.}\PY{p}{)}\PY{o}{*}\PY{n}{h}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{+} \PY{p}{(}\PY{l+m+mf}{1.}\PY{o}{/}\PY{l+m+mf}{12.}\PY{p}{)}\PY{o}{*}\PY{n}{h}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{xm}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{o}{*}\PY{n}{h} \PY{o}{+} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{h}\PY{o}{*}\PY{n}{xm}\PY{p}{)}\PY{p}{)}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{c+c1}{\PYZsh{} Numerical solution}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+se}{\PYZbs{}n}\PY{l+s+s2}{Numerical soltion to element matrix and element rhs: }\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{phi} \PY{o}{=} \PY{n}{basis}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)}
\PY{n}{A\PYZus{}e} \PY{o}{=} \PY{n}{element\PYZus{}matrix}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}
\PY{n}{b\PYZus{}e} \PY{o}{=} \PY{n}{element\PYZus{}rhs}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{f}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{A\PYZus{}e}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{b\PYZus{}e}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
Analytical soltion to element matrix and element rhs (only for d=1):
h: 1
A[i][i]: 0.3333333333333333
A[i][j]: 0.16666666666666666
b[0]: 0.08333333333333334
b[1]: 0.08333333333333331
Numerical soltion to element matrix and element rhs:
[[0.333334 0.166666]
[0.166666 0.333334]]
[0.083333 0.083333]
\end{Verbatim}
\subsection{Global matrix / Finite Element Assembly
Matrix}\label{global-matrix-finite-element-assembly-matrix}
The global matrix entries \(A_{i,j}\) are the sum of the local element
matrix entries \[
A_{i,j} = \int_{\Omega} \phi_i \phi_j \text{d}x = \sum_{e=0}^{N} A_{i,j}^{(e)}
\]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}50}]:} \PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline
\PY{k+kn}{from} \PY{n+nn}{matplotlib} \PY{k}{import} \PY{n}{pyplot} \PY{k}{as} \PY{n}{plt}
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Uniform\PYZus{}nodes}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{b}\PY{p}{,} \PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}Computes uniformly spaced nodes\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{h} \PY{o}{=} \PY{p}{(}\PY{n}{b}\PY{o}{\PYZhy{}}\PY{n}{a}\PY{p}{)} \PY{o}{/} \PY{n}{N}
\PY{k}{return} \PY{p}{[}\PY{n}{a} \PY{o}{+} \PY{n}{i}\PY{o}{*}\PY{n}{h} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{init\PYZus{}mesh}\PY{p}{(}\PY{n}{Omega}\PY{p}{,} \PY{n}{N}\PY{p}{,} \PY{n}{n}\PY{p}{,} \PY{o}{*}\PY{o}{*}\PY{n}{kwargs}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Function initializes a 2D array which defines N+1 global }
\PY{l+s+sd}{ 1D finite elements, with n+1 nodes inside each element}
\PY{l+s+sd}{ in the domain Omega = (x0, x1) \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{node\PYZus{}spacing} \PY{o}{=} \PY{n}{kwargs}\PY{o}{.}\PY{n}{get}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{node\PYZus{}spacing}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{uniform}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{if} \PY{p}{(}\PY{n}{node\PYZus{}spacing} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{uniform}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{:}
\PY{n}{nodes} \PY{o}{=} \PY{n}{Uniform\PYZus{}nodes}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{+} \PY{n}{N}\PY{o}{*}\PY{n}{n}\PY{p}{)}
\PY{k}{elif} \PY{p}{(}\PY{n}{node\PYZus{}spacing} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{chebyshev}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{:}
\PY{n}{nodes} \PY{o}{=} \PY{n}{Chebyshev\PYZus{}nodes}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{+} \PY{n}{N}\PY{o}{*}\PY{n}{n}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Initialize elements}
\PY{n}{elements} \PY{o}{=} \PY{p}{[}\PY{p}{[}\PY{n}{i}\PY{o}{\PYZhy{}}\PY{n}{j}\PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{j}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{n}{j}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{]} \PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{return} \PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{)}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{assemble}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{f}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Function to assemble the global linear system of }
\PY{l+s+sd}{ equations A*x=b for a given mesh which is defined }
\PY{l+s+sd}{ through nodes and elements}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{N\PYZus{}n}\PY{p}{,} \PY{n}{N\PYZus{}e} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{nodes}\PY{p}{)}\PY{p}{,} \PY{n+nb}{len}\PY{p}{(}\PY{n}{elements}\PY{p}{)}
\PY{n}{A} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{p}{(}\PY{n}{N\PYZus{}n}\PY{p}{,} \PY{n}{N\PYZus{}n}\PY{p}{)}\PY{p}{)}
\PY{n}{b} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{N\PYZus{}n}\PY{p}{)}
\PY{k}{for} \PY{n}{e} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N\PYZus{}e}\PY{p}{)}\PY{p}{:}
\PY{n}{Omega\PYZus{}e} \PY{o}{=} \PY{p}{(}\PY{n}{nodes}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]}\PY{p}{,} \PY{n}{nodes}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}\PY{p}{)}
\PY{n}{A\PYZus{}e} \PY{o}{=} \PY{n}{element\PYZus{}matrix}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}
\PY{n}{b\PYZus{}e} \PY{o}{=} \PY{n}{element\PYZus{}rhs}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{f}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{n}{A}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{p}{,} \PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{p}{]} \PY{o}{+}\PY{o}{=} \PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{,} \PY{n}{s}\PY{p}{]}
\PY{n}{b}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{p}{]} \PY{o}{+}\PY{o}{=} \PY{n}{b\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{]}
\PY{k}{return} \PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{u\PYZus{}glob}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{C}\PY{p}{,} \PY{n}{resolution}\PY{p}{)}\PY{p}{:}
\PY{n}{x\PYZus{}patches} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{u\PYZus{}patches} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{k}{for} \PY{n}{e} \PY{o+ow}{in} \PY{n}{elements}\PY{p}{:}
\PY{p}{(}\PY{n}{x\PYZus{}L}\PY{p}{,} \PY{n}{x\PYZus{}R}\PY{p}{)} \PY{o}{=} \PY{p}{(}\PY{n}{nodes}\PY{p}{[}\PY{n}{e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]}\PY{p}{,} \PY{n}{nodes}\PY{p}{[}\PY{n}{e}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}\PY{p}{)}
\PY{n}{d} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{e}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}
\PY{n}{X} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{resolution}\PY{p}{)}
\PY{n}{x} \PY{o}{=} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{p}{(}\PY{n}{x\PYZus{}L} \PY{o}{+} \PY{n}{x\PYZus{}R}\PY{p}{)} \PY{o}{+} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{p}{(}\PY{n}{x\PYZus{}R} \PY{o}{\PYZhy{}} \PY{n}{x\PYZus{}L}\PY{p}{)}\PY{o}{*}\PY{n}{X}
\PY{n}{x\PYZus{}patches}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{n}{u\PYZus{}element} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{resolution}\PY{p}{)}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{e}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{n}{i} \PY{o}{=} \PY{n}{e}\PY{p}{[}\PY{n}{r}\PY{p}{]} \PY{c+c1}{\PYZsh{} global node indice}
\PY{n}{u\PYZus{}element} \PY{o}{+}\PY{o}{=} \PY{n}{C}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{*} \PY{n}{phi\PYZus{}r}\PY{p}{(}\PY{n}{r}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)}
\PY{n}{u\PYZus{}patches}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{u\PYZus{}element}\PY{p}{)}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{n}{x\PYZus{}patches}\PY{p}{)}
\PY{n}{u} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{n}{u\PYZus{}patches}\PY{p}{)}
\PY{k}{return} \PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{u}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Test assembly of linear equation system}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k+kn}{from} \PY{n+nn}{numpy} \PY{k}{import} \PY{n}{tanh}\PY{p}{,} \PY{n}{pi}\PY{p}{,} \PY{n}{sin}\PY{p}{,} \PY{n}{cos}\PY{p}{,} \PY{n}{exp}
\PY{n}{d} \PY{o}{=} \PY{l+m+mi}{3}
\PY{n}{N} \PY{o}{=} \PY{l+m+mi}{5}
\PY{n}{Omega} \PY{o}{=} \PY{p}{(}\PY{o}{\PYZhy{}}\PY{n}{pi}\PY{p}{,} \PY{n}{pi}\PY{p}{)}
\PY{n}{f} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{:} \PY{o}{\PYZhy{}}\PY{n}{tanh}\PY{p}{(}\PY{n}{x}\PY{p}{)} \PY{o}{*} \PY{n}{sin}\PY{p}{(}\PY{n}{x}\PY{o}{\PYZhy{}}\PY{n}{pi}\PY{p}{)} \PY{o}{*} \PY{n}{exp}\PY{p}{(}\PY{l+m+mf}{1.} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.75}\PY{o}{*} \PY{p}{(}\PY{n}{x}\PY{o}{\PYZhy{}}\PY{l+m+mf}{0.25}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{)}
\PY{c+c1}{\PYZsh{}Omega = (0, 1)}
\PY{c+c1}{\PYZsh{}f = lambda x: x*(1\PYZhy{}x)**8}
\PY{n}{elem\PYZus{}resolution} \PY{o}{=} \PY{l+m+mi}{40}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{elem\PYZus{}resolution}\PY{p}{)}
\PY{n}{X} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{d}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{phi} \PY{o}{=} \PY{n}{basis}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Initialize mesh}
\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{)} \PY{o}{=} \PY{n}{init\PYZus{}mesh}\PY{p}{(}\PY{n}{Omega}\PY{p}{,} \PY{n}{N}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{n}{d}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Assemble the global linear equation system }
\PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)} \PY{o}{=} \PY{n}{assemble}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{f}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Solve the global linear equation system}
\PY{n}{C} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{solve}\PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Compute the approximation u(x) to f(x)}
\PY{p}{(}\PY{n}{x\PYZus{}u}\PY{p}{,} \PY{n}{u}\PY{p}{)} \PY{o}{=} \PY{n}{u\PYZus{}glob}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{C}\PY{p}{,} \PY{l+m+mi}{40}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Plot results}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{n}{element\PYZus{}edges} \PY{o}{=} \PY{p}{[}\PY{n}{nodes}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{]}
\PY{n}{element\PYZus{}edges}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{nodes}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{fig} \PY{o}{=} \PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
\PY{n}{fig}\PY{o}{.}\PY{n}{set\PYZus{}size\PYZus{}inches}\PY{p}{(}\PY{l+m+mi}{8}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{ax} \PY{o}{=} \PY{n}{fig}\PY{o}{.}\PY{n}{add\PYZus{}subplot}\PY{p}{(}\PY{l+m+mi}{111}\PY{p}{)}
\PY{n}{e\PYZus{}plt} \PY{o}{=} \PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{element\PYZus{}edges}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{*}\PY{n+nb}{len}\PY{p}{(}\PY{n}{element\PYZus{}edges}\PY{p}{)}\PY{p}{,} \PY{n}{ls}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{None}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{marker}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{s}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{f\PYZus{}plt} \PY{o}{=} \PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{f}\PY{p}{(}\PY{n}{x}\PY{p}{)}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{r}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{ls}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZdl{}f(x)\PYZdl{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{u\PYZus{}plt} \PY{o}{=} \PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x\PYZus{}u}\PY{p}{,} \PY{n}{u}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{k}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{ls}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZhy{}\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZdl{}u(x)\PYZdl{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{f(x), u(x)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_4_0.png}
\end{center}
{ \hspace*{\fill} \\}
\section{Finite Element Method in 1D using Collocation
Method}\label{finite-element-method-in-1d-using-collocation-method}
For the collocating finite element method, the entries in the element
matrices and element right hand sides are not evaluated through
integration. Instead the are calculated through: \[
\tilde{A}^{(e)}_{r,s} = \phi_r(x_s) \text{det}(J)
\] \[
\tilde{b}^{(e)}_{r} = f(x_r) \text{det}(J)
\] since \(\phi_r(x_s) = 1\) for \(r = s\) and \(0\) for all other
nodes.
Therefore, we do not need no longer the trapezoidal rule for the
numerical integration of terms like
\(\int_{\Omega^{(e)}} \phi_{q(e,r)} \phi_{q(e,s)} \text{d}x\).
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}1}]:} \PY{c+c1}{\PYZsh{} https://github.com/hplgit/INF5620/blob/master/src/fem/fe\PYZus{}approx1D.py}
\PY{o}{\PYZpc{}}\PY{k}{matplotlib} inline
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\PY{k+kn}{from} \PY{n+nn}{matplotlib} \PY{k}{import} \PY{n}{pyplot} \PY{k}{as} \PY{n}{plt}
\PY{c+c1}{\PYZsh{}=======================================================}
\PY{c+c1}{\PYZsh{} Implementation of the Finite Element Method in 1D}
\PY{c+c1}{\PYZsh{} for the approximateion of a function f(x)}
\PY{c+c1}{\PYZsh{}=======================================================}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Uniform\PYZus{}nodes}\PY{p}{(}\PY{n}{a}\PY{p}{,} \PY{n}{b}\PY{p}{,} \PY{n}{N}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{}Computes uniformly spaced nodes\PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{h} \PY{o}{=} \PY{p}{(}\PY{n}{b}\PY{o}{\PYZhy{}}\PY{n}{a}\PY{p}{)} \PY{o}{/} \PY{n}{N}
\PY{k}{return} \PY{p}{[}\PY{n}{a} \PY{o}{+} \PY{n}{i}\PY{o}{*}\PY{n}{h} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{init\PYZus{}mesh}\PY{p}{(}\PY{n}{Omega}\PY{p}{,} \PY{n}{N}\PY{p}{,} \PY{n}{n}\PY{p}{,} \PY{o}{*}\PY{o}{*}\PY{n}{kwargs}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Function initializes a 2D array which defines N+1 global }
\PY{l+s+sd}{ 1D finite elements, with n+1 nodes inside each element}
\PY{l+s+sd}{ in the domain Omega = (x0, x1) \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{node\PYZus{}spacing} \PY{o}{=} \PY{n}{kwargs}\PY{o}{.}\PY{n}{get}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{node\PYZus{}spacing}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{uniform}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{k}{if} \PY{p}{(}\PY{n}{node\PYZus{}spacing} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{uniform}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{:}
\PY{n}{nodes} \PY{o}{=} \PY{n}{Uniform\PYZus{}nodes}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{+} \PY{n}{N}\PY{o}{*}\PY{n}{n}\PY{p}{)}
\PY{k}{elif} \PY{p}{(}\PY{n}{node\PYZus{}spacing} \PY{o}{==} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{chebyshev}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{:}
\PY{n}{nodes} \PY{o}{=} \PY{n}{Chebyshev\PYZus{}nodes}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n} \PY{o}{+} \PY{n}{N}\PY{o}{*}\PY{n}{n}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Initialize elements}
\PY{n}{elements} \PY{o}{=} \PY{p}{[}\PY{p}{[}\PY{n}{i}\PY{o}{\PYZhy{}}\PY{n}{j}\PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{j}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{,} \PY{p}{(}\PY{n}{j}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{)}\PY{p}{]} \PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{return} \PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{)}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{Lagrange\PYZus{}polynomial}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{i}\PY{p}{,} \PY{n}{points}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Compute lagrangian interpolation polynomial }
\PY{l+s+sd}{ over domain x at specific points \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{p} \PY{o}{=} \PY{l+m+mi}{1}
\PY{k}{for} \PY{n}{k} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{points}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{k}{if} \PY{n}{k} \PY{o}{!=} \PY{n}{i}\PY{p}{:}
\PY{n}{p} \PY{o}{*}\PY{o}{=} \PY{p}{(}\PY{n}{x} \PY{o}{\PYZhy{}} \PY{n}{points}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{)} \PY{o}{/} \PY{p}{(}\PY{n}{points}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{points}\PY{p}{[}\PY{n}{k}\PY{p}{]}\PY{p}{)}
\PY{k}{return} \PY{n}{p}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{phi\PYZus{}r}\PY{p}{(}\PY{n}{r}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Defines a Lagrangian basis function for the }
\PY{l+s+sd}{ local node r of an reference element of }
\PY{l+s+sd}{ degree d \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{nodes} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mf}{1.}\PY{p}{,} \PY{l+m+mf}{1.}\PY{p}{,} \PY{n}{d}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}
\PY{k}{return} \PY{n}{Lagrange\PYZus{}polynomial}\PY{p}{(}\PY{n}{X}\PY{p}{,} \PY{n}{r}\PY{p}{,} \PY{n}{nodes}\PY{p}{)}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{basis}\PY{p}{(}\PY{n}{d}\PY{o}{=}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Create all basis functions for a reference }
\PY{l+s+sd}{ element of degree d}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{X} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{d}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{phi} \PY{o}{=} \PY{p}{[}\PY{n}{phi\PYZus{}r}\PY{p}{(}\PY{n}{r}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)} \PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{d}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}\PY{p}{]}
\PY{k}{return} \PY{n}{phi}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{element\PYZus{}matrix}\PY{p}{(}\PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Creates the element matrix for an element over}
\PY{l+s+sd}{ its local domain Omega\PYZus{}e=[x0\PYZus{}e, x1\PYZus{}e] and its }
\PY{l+s+sd}{ basis functions stored in phi }
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{n} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{phi}\PY{p}{)}
\PY{n}{A\PYZus{}e} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{p}{(}\PY{n}{n}\PY{p}{,}\PY{n}{n}\PY{p}{)}\PY{p}{)}
\PY{n}{h} \PY{o}{=} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{detJ} \PY{o}{=} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{h}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{,} \PY{n}{s}\PY{p}{]} \PY{o}{=} \PY{n}{phi}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{o}{*}\PY{n}{detJ}
\PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{s}\PY{p}{,} \PY{n}{r}\PY{p}{]} \PY{o}{=} \PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{,} \PY{n}{s}\PY{p}{]}
\PY{k}{return} \PY{n}{A\PYZus{}e}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{element\PYZus{}rhs}\PY{p}{(}\PY{n}{f}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}\PY{p}{:}
\PY{n}{n} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{phi}\PY{p}{)}
\PY{n}{b\PYZus{}e} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{n}\PY{p}{)}
\PY{n}{h} \PY{o}{=} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{detJ} \PY{o}{=} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{n}{h}
\PY{c+c1}{\PYZsh{} Compute mapping from f(x) to f(x(X))}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{n}\PY{p}{)}
\PY{n}{f\PYZus{}map} \PY{o}{=} \PY{n}{f}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{p}{:}
\PY{n}{b\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{]} \PY{o}{=} \PY{n}{f\PYZus{}map}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{o}{*}\PY{n}{detJ}
\PY{k}{return} \PY{n}{b\PYZus{}e}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k}{def} \PY{n+nf}{assemble}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{f}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZsq{}\PYZsq{}\PYZsq{} Function to assemble the global linear system of }
\PY{l+s+sd}{ equations A*x=b for a given mesh which is defined }
\PY{l+s+sd}{ through nodes and elements}
\PY{l+s+sd}{ \PYZsq{}\PYZsq{}\PYZsq{}}
\PY{n}{N\PYZus{}n}\PY{p}{,} \PY{n}{N\PYZus{}e} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{nodes}\PY{p}{)}\PY{p}{,} \PY{n+nb}{len}\PY{p}{(}\PY{n}{elements}\PY{p}{)}
\PY{n}{A} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{p}{(}\PY{n}{N\PYZus{}n}\PY{p}{,} \PY{n}{N\PYZus{}n}\PY{p}{)}\PY{p}{)}
\PY{n}{b} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{N\PYZus{}n}\PY{p}{)}
\PY{k}{for} \PY{n}{e} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n}{N\PYZus{}e}\PY{p}{)}\PY{p}{:}
\PY{n}{Omega\PYZus{}e} \PY{o}{=} \PY{p}{(}\PY{n}{nodes}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]}\PY{p}{,} \PY{n}{nodes}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}\PY{p}{)}
\PY{n}{A\PYZus{}e} \PY{o}{=} \PY{n}{element\PYZus{}matrix}\PY{p}{(}\PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}
\PY{n}{b\PYZus{}e} \PY{o}{=} \PY{n}{element\PYZus{}rhs}\PY{p}{(}\PY{n}{f}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{Omega\PYZus{}e}\PY{p}{)}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{k}{for} \PY{n}{s} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{n}{A}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{p}{,} \PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{n}{s}\PY{p}{]}\PY{p}{]} \PY{o}{+}\PY{o}{=} \PY{n}{A\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{,} \PY{n}{s}\PY{p}{]}
\PY{n}{b}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{e}\PY{p}{]}\PY{p}{[}\PY{n}{r}\PY{p}{]}\PY{p}{]} \PY{o}{+}\PY{o}{=} \PY{n}{b\PYZus{}e}\PY{p}{[}\PY{n}{r}\PY{p}{]}
\PY{k}{return} \PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{u\PYZus{}glob}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{C}\PY{p}{,} \PY{n}{resolution}\PY{p}{)}\PY{p}{:}
\PY{n}{x\PYZus{}patches} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{n}{u\PYZus{}patches} \PY{o}{=} \PY{p}{[}\PY{p}{]}
\PY{k}{for} \PY{n}{e} \PY{o+ow}{in} \PY{n}{elements}\PY{p}{:}
\PY{p}{(}\PY{n}{x\PYZus{}L}\PY{p}{,} \PY{n}{x\PYZus{}R}\PY{p}{)} \PY{o}{=} \PY{p}{(}\PY{n}{nodes}\PY{p}{[}\PY{n}{e}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]}\PY{p}{,} \PY{n}{nodes}\PY{p}{[}\PY{n}{e}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}\PY{p}{)}
\PY{n}{d} \PY{o}{=} \PY{n+nb}{len}\PY{p}{(}\PY{n}{e}\PY{p}{)}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}
\PY{n}{X} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{,} \PY{n}{resolution}\PY{p}{)}
\PY{n}{x} \PY{o}{=} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{p}{(}\PY{n}{x\PYZus{}L} \PY{o}{+} \PY{n}{x\PYZus{}R}\PY{p}{)} \PY{o}{+} \PY{l+m+mf}{0.5}\PY{o}{*}\PY{p}{(}\PY{n}{x\PYZus{}R} \PY{o}{\PYZhy{}} \PY{n}{x\PYZus{}L}\PY{p}{)}\PY{o}{*}\PY{n}{X}
\PY{n}{x\PYZus{}patches}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\PY{n}{u\PYZus{}element} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros}\PY{p}{(}\PY{n}{resolution}\PY{p}{)}
\PY{k}{for} \PY{n}{r} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{e}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{n}{i} \PY{o}{=} \PY{n}{e}\PY{p}{[}\PY{n}{r}\PY{p}{]} \PY{c+c1}{\PYZsh{} global node indice}
\PY{n}{u\PYZus{}element} \PY{o}{+}\PY{o}{=} \PY{n}{C}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{*} \PY{n}{phi\PYZus{}r}\PY{p}{(}\PY{n}{r}\PY{p}{,} \PY{n}{X}\PY{p}{,} \PY{n}{d}\PY{p}{)}
\PY{n}{u\PYZus{}patches}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{u\PYZus{}element}\PY{p}{)}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{n}{x\PYZus{}patches}\PY{p}{)}
\PY{n}{u} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{concatenate}\PY{p}{(}\PY{n}{u\PYZus{}patches}\PY{p}{)}
\PY{k}{return} \PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{u}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Test assembly of linear equation system}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{k+kn}{from} \PY{n+nn}{numpy} \PY{k}{import} \PY{n}{tanh}\PY{p}{,} \PY{n}{pi}\PY{p}{,} \PY{n}{sin}\PY{p}{,} \PY{n}{cos}\PY{p}{,} \PY{n}{exp}
\PY{n}{d} \PY{o}{=} \PY{l+m+mi}{2}
\PY{n}{N} \PY{o}{=} \PY{l+m+mi}{7}
\PY{n}{Omega} \PY{o}{=} \PY{p}{(}\PY{o}{\PYZhy{}}\PY{n}{pi}\PY{p}{,} \PY{n}{pi}\PY{p}{)}
\PY{n}{f} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{:} \PY{o}{\PYZhy{}}\PY{n}{tanh}\PY{p}{(}\PY{n}{x}\PY{p}{)} \PY{o}{*} \PY{n}{sin}\PY{p}{(}\PY{n}{x}\PY{o}{\PYZhy{}}\PY{n}{pi}\PY{p}{)} \PY{o}{*} \PY{n}{exp}\PY{p}{(}\PY{l+m+mf}{1.} \PY{o}{\PYZhy{}} \PY{l+m+mf}{0.75}\PY{o}{*} \PY{p}{(}\PY{n}{x}\PY{o}{\PYZhy{}}\PY{l+m+mf}{0.25}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{)}
\PY{c+c1}{\PYZsh{}Omega = (0, 1)}
\PY{c+c1}{\PYZsh{}f = lambda x: x*(1\PYZhy{}x)**8}
\PY{n}{elem\PYZus{}resolution} \PY{o}{=} \PY{l+m+mi}{40}
\PY{n}{x} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,} \PY{n}{Omega}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,} \PY{n}{elem\PYZus{}resolution}\PY{p}{)}
\PY{n}{phi} \PY{o}{=} \PY{n}{basis}\PY{p}{(}\PY{n}{d}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Initialize mesh}
\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{)} \PY{o}{=} \PY{n}{init\PYZus{}mesh}\PY{p}{(}\PY{n}{Omega}\PY{p}{,} \PY{n}{N}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{,} \PY{n}{d}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Assemble the global linear equation system }
\PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)} \PY{o}{=} \PY{n}{assemble}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{phi}\PY{p}{,} \PY{n}{f}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Solve the global linear equation system}
\PY{n}{C} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{solve}\PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Compute the approximation u(x) to f(x)}
\PY{p}{(}\PY{n}{x\PYZus{}u}\PY{p}{,} \PY{n}{u}\PY{p}{)} \PY{o}{=} \PY{n}{u\PYZus{}glob}\PY{p}{(}\PY{n}{nodes}\PY{p}{,} \PY{n}{elements}\PY{p}{,} \PY{n}{C}\PY{p}{,} \PY{l+m+mi}{40}\PY{p}{)}
\PY{c+c1}{\PYZsh{} Plot results}
\PY{c+c1}{\PYZsh{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}\PYZhy{}}
\PY{n}{element\PYZus{}edges} \PY{o}{=} \PY{p}{[}\PY{n}{nodes}\PY{p}{[}\PY{n}{elements}\PY{p}{[}\PY{n}{i}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{]} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,}\PY{n}{N}\PY{p}{)}\PY{p}{]}
\PY{n}{element\PYZus{}edges}\PY{o}{.}\PY{n}{append}\PY{p}{(}\PY{n}{nodes}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{fig} \PY{o}{=} \PY{n}{plt}\PY{o}{.}\PY{n}{figure}\PY{p}{(}\PY{p}{)}
\PY{n}{fig}\PY{o}{.}\PY{n}{set\PYZus{}size\PYZus{}inches}\PY{p}{(}\PY{l+m+mi}{8}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{)}
\PY{n}{ax} \PY{o}{=} \PY{n}{fig}\PY{o}{.}\PY{n}{add\PYZus{}subplot}\PY{p}{(}\PY{l+m+mi}{111}\PY{p}{)}
\PY{n}{e\PYZus{}plt} \PY{o}{=} \PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{element\PYZus{}edges}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{*}\PY{n+nb}{len}\PY{p}{(}\PY{n}{element\PYZus{}edges}\PY{p}{)}\PY{p}{,} \PY{n}{ls}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{None}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{marker}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{s}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{f\PYZus{}plt} \PY{o}{=} \PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x}\PY{p}{,} \PY{n}{f}\PY{p}{(}\PY{n}{x}\PY{p}{)}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{r}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{ls}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZdl{}f(x)\PYZdl{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{u\PYZus{}plt} \PY{o}{=} \PY{n}{ax}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{x\PYZus{}u}\PY{p}{,} \PY{n}{u}\PY{p}{,} \PY{n}{c}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{k}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{ls}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZhy{}\PYZhy{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{,} \PY{n}{label}\PY{o}{=}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{\PYZdl{}u(x)\PYZdl{}}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}xlabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{x}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{set\PYZus{}ylabel}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{f(x), u(x)}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}
\PY{n}{ax}\PY{o}{.}\PY{n}{legend}\PY{p}{(}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{show}\PY{p}{(}\PY{p}{)}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_6_0.png}
\end{center}
{ \hspace*{\fill} \\}
% Add a bibliography block to the postdoc
\end{document}