Finite Element Method for u''(x) + f(x) = 0.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x): # function f\n",
    "    return np.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 0 \n",
    "b = 1 # limits of x\n",
    "\n",
    "n = 100 # number of partitions\n",
    "\n",
    "dx = 1/n\n",
    "Xmatrix = np.linspace(0, 1, n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Boundary conditions on $u$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_a = 0\n",
    "u_b = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We want to solve for $U$ in $KU=F$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Necessary functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$F_j = \\int_{0}^{1}f(x)N_{j}(x)dx$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "def init_Fmatrix():  # function for generating matrix F\n",
    "    Fmatrix = np.zeros((n,1))\n",
    "    for j in range(n):\n",
    "        x_j = Xmatrix[j]\n",
    "        Fmatrix[j] = f(x_j)*dx\n",
    "    Fmatrix[0] += u_a/dx\n",
    "    Fmatrix[-1] += u_b/dx\n",
    "    return Fmatrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$K_{ij} = \\int_{0}^{1}N'_{i}(x)N'_{j}(x)dx$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "def init_Kmatrix():  # function for generating matrix K\n",
    "    Kmatrix = np.zeros((n,n))\n",
    "    for i in range(n):\n",
    "        for j in range(i, n):\n",
    "            if i == j:\n",
    "                Kmatrix[i,j] = 2/dx\n",
    "            \n",
    "            if j - i == 1:\n",
    "                Kmatrix[i,j] = -1/dx\n",
    "                Kmatrix[j,i] = -1/dx\n",
    "    return Kmatrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrix K"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solving for U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fef660be190>]"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Fmatrix = init_Fmatrix()\n",
    "Kmatrix = init_Kmatrix()\n",
    "\n",
    "Umatrix = np.linalg.solve(Kmatrix, Fmatrix)\n",
    "\n",
    "plt.plot(Xmatrix, Umatrix, lw=0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.00158368],\n",
       "       [0.00316737],\n",
       "       [0.00475004],\n",
       "       [0.00633069],\n",
       "       [0.00790831],\n",
       "       [0.0094819 ],\n",
       "       [0.01105043],\n",
       "       [0.01261291],\n",
       "       [0.01416833],\n",
       "       [0.01571567],\n",
       "       [0.01725393],\n",
       "       [0.01878211],\n",
       "       [0.0202992 ],\n",
       "       [0.0218042 ],\n",
       "       [0.02329611],\n",
       "       [0.02477392],\n",
       "       [0.02623664],\n",
       "       [0.02768326],\n",
       "       [0.0291128 ],\n",
       "       [0.03052426],\n",
       "       [0.03191664],\n",
       "       [0.03328896],\n",
       "       [0.03464023],\n",
       "       [0.03596945],\n",
       "       [0.03727565],\n",
       "       [0.03855785],\n",
       "       [0.03981506],\n",
       "       [0.04104631],\n",
       "       [0.04225062],\n",
       "       [0.04342703],\n",
       "       [0.04457456],\n",
       "       [0.04569224],\n",
       "       [0.04677913],\n",
       "       [0.04783425],\n",
       "       [0.04885665],\n",
       "       [0.04984538],\n",
       "       [0.05079949],\n",
       "       [0.05171803],\n",
       "       [0.05260006],\n",
       "       [0.05344464],\n",
       "       [0.05425084],\n",
       "       [0.05501773],\n",
       "       [0.05574437],\n",
       "       [0.05642985],\n",
       "       [0.05707325],\n",
       "       [0.05767366],\n",
       "       [0.05823016],\n",
       "       [0.05874185],\n",
       "       [0.05920783],\n",
       "       [0.0596272 ],\n",
       "       [0.05999907],\n",
       "       [0.06032255],\n",
       "       [0.06059677],\n",
       "       [0.06082085],\n",
       "       [0.06099391],\n",
       "       [0.06111509],\n",
       "       [0.06118353],\n",
       "       [0.06119838],\n",
       "       [0.06115877],\n",
       "       [0.06106388],\n",
       "       [0.06091285],\n",
       "       [0.06070486],\n",
       "       [0.06043908],\n",
       "       [0.06011469],\n",
       "       [0.05973087],\n",
       "       [0.05928681],\n",
       "       [0.05878171],\n",
       "       [0.05821478],\n",
       "       [0.05758522],\n",
       "       [0.05689225],\n",
       "       [0.05613508],\n",
       "       [0.05531296],\n",
       "       [0.05442511],\n",
       "       [0.05347078],\n",
       "       [0.05244921],\n",
       "       [0.05135966],\n",
       "       [0.0502014 ],\n",
       "       [0.04897369],\n",
       "       [0.04767581],\n",
       "       [0.04630705],\n",
       "       [0.04486669],\n",
       "       [0.04335403],\n",
       "       [0.04176839],\n",
       "       [0.04010906],\n",
       "       [0.03837538],\n",
       "       [0.03656668],\n",
       "       [0.03468228],\n",
       "       [0.03272153],\n",
       "       [0.03068379],\n",
       "       [0.0285684 ],\n",
       "       [0.02637475],\n",
       "       [0.02410221],\n",
       "       [0.02175015],\n",
       "       [0.01931797],\n",
       "       [0.01680508],\n",
       "       [0.01421087],\n",
       "       [0.01153476],\n",
       "       [0.00877619],\n",
       "       [0.00593457],\n",
       "       [0.00300936]])"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Umatrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}