FDM_FEM_1D.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "interpolation_projection.ipynb",
      "version": "0.3.2",
      "provenance": [],
      "collapsed_sections": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/van-dang/FEniCS-Colab/blob/master/FDM_FEM_1D.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "metadata": {
        "id": "DIqQdid0Sawk",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# Finite Difference and Finite Element Method for 1D Poisson\n",
        "\n",
        "After this exercise, students should be able to\n",
        "\n",
        "*   Understand the difference between two numerical methods to solve a PDE: Finite Different Method and FInite Element Method.\n",
        "*   Explain two concepts: strong and weak solutions of a PDE.\n",
        "*   Bring two methods to practice.\n",
        "\n",
        "\n",
        "Consider 1D Poisson\n",
        "$$-u''=f\\,\\qquad x\\in(a, b)$$\n",
        "with Dirichlet boundary conditions $u(a)=u_a$ and $u(b)=u_b$\n",
        "where $u_a$ and $u_b$ are given.\n",
        "\n",
        "We will find the strong solution of the equation by the Finite Difference Method and weak solution by the Finite Element Method.\n",
        "\n",
        "First, we discretize the domain $[a, b]$ into $N-1$ subintervals by $N$ grid points $\\{x_k\\}_{k=0..N-1}$, $x_0=a$ and $x_{N-1}=b$, $\\Delta x=x_{k}-x_{k-1}$. A uniform discretization is used here for the simplicity and assume that  $u^k\\approx u(x_k)$. \n"
      ]
    },
    {
      "metadata": {
        "id": "iMMaYbjYZKKI",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "**Finite Difference Method** (FDM) uses the appproximation\n",
        "$$u''(x_k)\\approx\\frac{u^{k+1}-2u^{k}+u^{k-1}}{h^2}$$\n",
        "The discretized equation\n",
        "$$-\\frac{u^{k+1}-2u^{k}+u^{k-1}}{h^2}=f^k, \\qquad k=1..N-2$$\n",
        "Matrix form\n",
        "$$-\\frac{1}{h^2}\\begin{bmatrix} - & - &  & \\\\  1 & -2 & 1\\\\ & 1 & -2 & 1  \\\\&&&...\\\\ & &  & 1 & -2 & 1 \\\\ & & &  & 1 & -2 & 1 \\\\ & & &  &  & - & - \n",
        "\\end{bmatrix}\\begin{bmatrix}  u^0 \\\\ u^1 \\\\ u^2 \\\\ ... \\\\ u^{N-3}\\\\ u^{N-2} \\\\ u^{N-1}\\end{bmatrix}  = \\begin{bmatrix}  f^0 \\\\ f^1 \\\\ f^2 \\\\ ... \\\\ f^{N-3}\\\\ f^{N-2} \\\\ f^{N-1}\\end{bmatrix}$$\n",
        "\n",
        "Apply the Dirichlet boundary conditions $u^0=u_a, u^{N-1}=u_b$\n",
        "\n",
        "$$\\begin{bmatrix} 1 &  &  & \\\\  -\\frac{1}{h^2} & \\frac{2}{h^2} & -\\frac{1}{h^2}\\\\ & -\\frac{1}{h^2} & \\frac{2}{h^2} & -\\frac{1}{h^2}  \\\\&&&...\\\\ & &  & -\\frac{1}{h^2} & \\frac{2}{h^2} & -\\frac{1}{h^2} \\\\ & & &  & -\\frac{1}{h^2} & \\frac{2}{h^2} & -\\frac{1}{h^2} \\\\ & & &  &  &  & 1 \n",
        "\\end{bmatrix}\\begin{bmatrix}  u^0 \\\\ u^1 \\\\ u^2 \\\\ ... \\\\ u^{N-3}\\\\ u^{N-2} \\\\ u^{N-1}\\end{bmatrix}  = \\begin{bmatrix}  u_a \\\\ f^1 \\\\ f^2 \\\\ ... \\\\ f^{N-3}\\\\ f^{N-2}\\\\u_b\\end{bmatrix}$$\n",
        "\n",
        "**Exercise**: Understand the code below."
      ]
    },
    {
      "metadata": {
        "id": "QaL0dgBMSbiq",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def FDM(f, a, b, N, u_bc):\n",
        "# Solving -u''=f in (a, b) by Finite Difference Method \n",
        "# Dirichlet boundary conditions at two ending points\n",
        "# Van-Dang Nguyen 2019  \n",
        "  xd= np.linspace(a, b, num=N)\n",
        "  # Grid size\n",
        "  h=xd[1]-xd[0]\n",
        "  ones=np.ones(N)\n",
        "\n",
        "  # Create left-hand side digonal matrix [1 -2 1]\n",
        "  data = -1/(h*h)*np.array([ones,-2*ones,ones])\n",
        "  diags = np.array([-1, 0, 1])\n",
        "  A=spdiags(data, diags, N, N,format=\"csr\").toarray()\n",
        "   \n",
        "  # Boundary conditions for the left-hand side \n",
        "  A[0,:] = 0; # zero out row 0 corresponding to x=a.\n",
        "  A[0,0]=1\n",
        "  A[N-1,:]=0; # zero out row N-1 corresponding to x=b.\n",
        "  A[N-1,N-1]=1\n",
        "  \n",
        "  # Boundary conditions for the right-hand side  \n",
        "  lam_f = sp.lambdify(x, f, modules=['numpy'])\n",
        "  rhs = lam_f(xd); # rhs = source \n",
        "  rhs[0], rhs[N-1]=u_bc(xd[0]), u_bc(xd[N-1]) # Dirichlet values\n",
        "  \n",
        "  ## solving the linear system\n",
        "  u= spsolve(A, rhs, permc_spec=None, use_umfpack=True)\n",
        "  return xd, u"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "KStGW6DKfNZA",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "**Finite Element Method** gives the weak solutions. Here we illustrate how to find the finite solution of the equation in a linear space.\n",
        "\n",
        "On each element (sub-interval) $[x_k, x_{k+1}]$ with $h_k=|x_{k+1}-x_k|$, $u$ can be approximated by  a linear function\n",
        "\n",
        "$u(x)\\approx -\\frac{1}{h_k}u^k(x-x_{k+1})+\\frac{1}{h_k}u^{k+1}(x-x_k)=u^k\\varphi_{k,0}(x)+u^{k+1}\\varphi_{k,1}(x)$\n",
        "\n",
        "$\\varphi_{k}(x)$ is known as a linear basis function in FEM literature, \n",
        "\n",
        "$$\n",
        "\\varphi_{k,0}(x)=-\\frac{1}{h_k}(x-x_{k+1}) , \\quad \\varphi_{k,1}(x)=\\frac{1}{h_k}(x-x_{k}) \n",
        "$$\n",
        "\n",
        "Multiply both sides of the equation with a test function $v$, and integrate \n",
        "\n",
        "$$\\int_{x_k}^{x_{k+1}} -u'' v\\, dx = \\int_{x_k}^{k+1} f v \\,dx$$\n",
        "\n",
        "$$-u'v\\,\\biggl|_{x^k}^{x_{k+1}}+\\int_{x_k}^{x_{k+1}} u' v'\\, dx = \\int_{x_k}^{x_{k+1}} f \\,v \\,dx$$\n",
        "\n",
        "Let $q_k=u'(x_k)\\,v(x_k)$, we have \n",
        "$$-q_{k+1}+q_{k}+u^k\\int_{x_k}^{x_{k+1}} \\varphi_{k,0}'\\, v'\\, dx +u^{k+1}\\int_{x_k}^{x_{k+1}} \\varphi_{k,1}'\\, v'\\, dx = \\int_{x_k}^{x_{k+1}} f \\,v \\,dx$$\n",
        "\n",
        "It is enough to test with $v=\\varphi_{k,j}, \\quad j=0,1$\n",
        "\n",
        "* $j=0$,\n",
        "$$u^k\\int_{x_k}^{x_{k+1}} \\varphi_{k,0}'\\, \\varphi_{k,0}'\\, dx +u^{k+1}\\int_{x_k}^{x_{k+1}} \\varphi_{k,1}'\\, \\varphi_{k,0}'\\, dx = \\int_{x_k}^{x_{k+1}} f \\,\\varphi_{k,0} \\,dx-q_{k}$$\n",
        "\n",
        "* $j=1$, \n",
        "$$u^k\\int_{x_k}^{x_{k+1}} \\varphi_{k,0}'\\, \\varphi_{k,1}'\\, dx +u^{k+1}\\int_{x_k}^{x_{k+1}} \\varphi_{k,1}'\\, \\varphi_{k,1}'\\, dx = \\int_{x_k}^{x_{k+1}} f \\,\\varphi_{k,1} \\,dx+q_{k+1}$$\n",
        "\n",
        "Matrix form\n",
        "$$A^k U^k = F^k + Q^k$$\n",
        "where\n",
        "$$A_{mj}^k=\\int_{x_k}^{x_{k+1}} \\varphi_{k,m}'\\, \\varphi_{k,j}'\\, dx,\\quad F_j^k=\\int_{x_k}^{x_{k+1}} f \\,\\varphi_{k,j} \\,dx$$\n",
        "\n",
        "Matrix assembly\n",
        "$$A U = F + Q$$\n",
        "\n",
        "$$\\begin{bmatrix} A^0_{00} & A^0_{10} \\\\ A^0_{01} & A^0_{11} + A^1_{00} & A^1_{10} \\\\ & \\,\\, A^1_{01} & A^1_{11}+A^2_{00} & A^2_{10}\\\\ & & &...\\\\\n",
        "& &A^{N-4}_{01} & A^{N-4}_{11}+A^{N-3}_{00} & A^{N-3}_{10} \\\\ & & &  A^{N-3}_{01} & A^{N-3}_{11} + A^{N-2}_{00} & A^{N-2}_{10} \\\\ & & && A^{N-2}_{01} & A^{N-2}_{11}\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}  u^0 \\\\ u^1 \\\\ u^2 \\\\ ... \\\\ u^{N-3}\\\\ u^{N-2} \\\\ u^{N-1}\\end{bmatrix}  = \\begin{bmatrix}  f^0 \\\\ f^1 \\\\ f^2 \\\\ ... \\\\ f^{N-3}\\\\ f^{N-2}\\\\f^{N-1}\\end{bmatrix}\n",
        "+\\begin{bmatrix}  -q_0 \\\\ -q_1+q_1 \\\\-q_2+q_2 \\\\ ... \\\\ -q_{N-3}+q_{N-3}\\\\ -q_{N-2}+q_{N-2}\\\\q_{N-1}\\end{bmatrix}\n",
        "$$\n",
        "\n",
        "Apply the Dirichlet boundary conditions $u^0=u_a, u^{N-1}=u_b$\n",
        "\n",
        "$$\\begin{bmatrix} 1 &  \\\\ A^0_{10} & A^0_{11} + A^1_{00} & A^1_{01} \\\\ & A^1_{10} & A^1_{11}+A^2_{00} & A^2_{01}\\\\ & & &...\\\\\n",
        "& &A^{N-4}_{10}  & A^{N-4}_{11}+A^{N-3}_{00} & A^{N-3}_{01} \\\\ & & &  A^{N-3}_{10} & A^{N-3}_{11} + A^{N-2}_{00} & A^{N-2}_{01} \\\\ &&&& & 1\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}  u^0 \\\\ u^1 \\\\ u^2 \\\\ ... \\\\ u^{N-3}\\\\ u^{N-2} \\\\ u^{N-1}\\end{bmatrix}  = \\begin{bmatrix}  u_a \\\\ f^1 \\\\ f^2 \\\\ ... \\\\ f^{N-3}\\\\ f^{N-2}\\\\u_b\\end{bmatrix}\n",
        "$$\n",
        "\n",
        "Explicitly, we have\n",
        "$$A_{00}^k=\\frac{1}{h_k}, \\quad A_{01}^k=A_{10}^k=-\\frac{1}{h_k},\\quad A_{11}^k=\\frac{1}{h_k}$$\n",
        "and the matrix form is\n",
        "\n",
        "$$\\begin{bmatrix} 1 &  \\\\ \\frac{-1}{h} & \\frac{2}{h} & \\frac{-1}{h} \\\\ & \\frac{-1}{h} & \\frac{2}{h} & \\frac{-1}{h}\\\\ & & &...\\\\\n",
        "& &\\frac{-1}{h} & \\frac{2}{h} & \\frac{-1}{h} \\\\ & & & \\frac{-1}{h} & \\frac{2}{h} & \\frac{-1}{h} \\\\ &&&& & 1\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}  u^0 \\\\ u^1 \\\\ u^2 \\\\ ... \\\\ u^{N-3}\\\\ u^{N-2} \\\\ u^{N-1}\\end{bmatrix}  = \\begin{bmatrix}  u_a \\\\ f^1 \\\\ f^2 \\\\ ... \\\\ f^{N-3}\\\\ f^{N-2}\\\\u_b\\end{bmatrix}\n",
        "$$\n",
        "\n",
        "**Exercise**: Understand the code below."
      ]
    },
    {
      "metadata": {
        "id": "9f9M8lH1L_uc",
        "colab_type": "code",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "def FEM(f, a, b, N, u_bc):\n",
        "# Solving -u''=f in (a, b) by Linear Finite Element Method\n",
        "# Dirichlet boundary conditions at two ending points\n",
        "# Van-Dang Nguyen 2019  \n",
        "  \n",
        "  xd= np.linspace(a, b, num=N)\n",
        "  # A = sps.coo_matrix((N, N)) # empty matrix\n",
        "  # rhs = sps.coo_matrix((N, 1)) # empty matrix\n",
        "  A = np.zeros( (N, N) )\n",
        "  rhs = [0]*N;\n",
        "  for ele in range(0, N-1):\n",
        "    node0, node1 = ele, ele+1;\n",
        "    x0, x1 = xd[node0], xd[node1]\n",
        "    phi = basis_func(x0, x1)\n",
        "    dphi = diff_basis_func(x0, x1)\n",
        "    b0 = sp.integrate(phi[0]*f, (x, x0, x1))\n",
        "    b1 = sp.integrate(phi[1]*f, (x, x0, x1))\n",
        "        \n",
        "    A00 = sp.integrate(dphi[0]*dphi[0], (x, x0, x1))\n",
        "    A01 = sp.integrate(dphi[0]*dphi[1], (x, x0, x1))\n",
        "    A10 = sp.integrate(dphi[1]*dphi[0], (x, x0, x1))\n",
        "    A11 = sp.integrate(dphi[1]*dphi[1], (x, x0, x1))\n",
        "    \n",
        "    A[node0, node0] += A00\n",
        "    A[node0, node1] += A01\n",
        "    A[node1, node0] += A10\n",
        "    A[node1, node1] += A11\n",
        "\n",
        "    \n",
        "    rhs[node0] += float(b0);\n",
        "    rhs[node1] += float(b1);\n",
        "\n",
        "  # Boundary conditions for the left-hand side \n",
        "  A[0,:] = 0; # zero out row 0 corresponding to x=a.\n",
        "  A[0,0]=1\n",
        "  A[N-1,:]=0; # zero out row N-1 corresponding to x=b.\n",
        "  A[N-1,N-1]=1\n",
        " \n",
        "  # Boundary conditions for the right-hand side  \n",
        "  rhs[0], rhs[N-1]=float(u_bc(xd[0])), float(u_bc(xd[N-1])) # Dirichlet values\n",
        "\n",
        "  ## solving the linear system\n",
        "  u= spsolve(A, rhs, permc_spec=None, use_umfpack=True)\n",
        "  return xd, u\n",
        "\n",
        "def basis_func(x0, x1):\n",
        "  L = x1 - x0\n",
        "  phi = [0] * 2\n",
        "  phi[0] = -(x-x1)/L\n",
        "  phi[1] = (x-x0)/L  \n",
        "  return phi\n",
        "\n",
        "def diff_basis_func(x0, x1):\n",
        "  L = x1 - x0\n",
        "  dphi = [0] * 2\n",
        "  dphi[0] = -1./L\n",
        "  dphi[1] = 1./L  \n",
        "  return dphi"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "wx2CoAfFdURW",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "# A comparison"
      ]
    },
    {
      "metadata": {
        "id": "7nrdltm8m5L8",
        "colab_type": "code",
        "outputId": "edb74b5a-8b88-4932-ba7a-681c27fc43d7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 398
        }
      },
      "cell_type": "code",
      "source": [
        "import sympy as sp\n",
        "import numpy as np\n",
        "from scipy.sparse import spdiags\n",
        "import matplotlib.pyplot as plt\n",
        "from scipy.sparse.linalg import spsolve\n",
        "\n",
        "x = sp.Symbol('x') # global variable\n",
        "\n",
        "\n",
        "# N discrete points of [a, b]\n",
        "a, b, N = -5.0, 5.0, 10; \n",
        "\n",
        "# Compute the source based on a given exact solution\n",
        "x = sp.Symbol('x')\n",
        "ue = sp.sin(x)+ sp.cos(x)\n",
        "\n",
        "f = -sp.diff(ue, x, 2);\n",
        "lam_ue = sp.lambdify(x, ue, modules=['numpy'])\n",
        "      \n",
        "xd, u_fem = FEM(f, a, b, N, lam_ue)  \n",
        "\n",
        "xd, u_fdm = FDM(f, a, b, N, lam_ue)  \n",
        "\n",
        "xe = np.linspace(a, b, 100)\n",
        "ue_fine = lam_ue(xe)\n",
        "\n",
        "plt.plot(xe, ue_fine,'-',xd, u_fem,'--o',xd, u_fdm,'-.s')\n",
        "plt.gca().legend(('Exact','FEM','FDM'))"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "/usr/local/lib/python3.6/dist-packages/scipy/sparse/linalg/dsolve/linsolve.py:133: SparseEfficiencyWarning: spsolve requires A be CSC or CSR matrix format\n",
            "  SparseEfficiencyWarning)\n"
          ],
          "name": "stderr"
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f4b0e0923c8>"
            ]
          },
          "metadata": {
            "tags": []
          },
          "execution_count": 3
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAAecAAAFKCAYAAAAnj5dkAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nCIKDEcVZEARBEByMKM6CIAiC4GD+P8p8nhncHksqAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 576x396 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "y1UIbqwUqOKA",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "Questions\n",
        "\n",
        "*   Manipulate the number of grid points $N$ to compare the accuracy between the two methods\n",
        "*   Manipulate the source $f$ (by changing the exact solution $u_e$) to compare the accuracy.\n",
        "*   Modify the code for the quadratic basis functions.\n",
        "*   Modify the code to solve $-u''+u = f$."
      ]
    }
  ]
}