p16.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p16: Poisson equation on [-1,1]x[-1,1] with u=0 on boundary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We solve the following Poisson problem\n",
    "$$\n",
    "u_{xx} + u_{yy} = 10\\sin(8x(y-1)), \\qquad -1 < x,y < 1, \\qquad u=0 \\quad \\mbox{on boundary}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from chebPy import cheb\n",
    "from numpy import meshgrid,sin,dot,eye,kron,zeros,reshape,linspace\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.pyplot import figure,subplot,plot,title,axis,xlabel,ylabel,spy\n",
    "from matplotlib import cm\n",
    "from scipy.linalg import solve\n",
    "from scipy.interpolate import interp2d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p6d0e6ab2d1\">\n",
       "   <rect height=\"434.88\" width=\"434.88\" x=\"7.2\" y=\"23.304375\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "N = 24; D,x = cheb(N); y = x;\n",
    "xx,yy = meshgrid(x[1:N],y[1:N])\n",
    "xx = reshape(xx,(N-1)**2)\n",
    "yy = reshape(yy,(N-1)**2)\n",
    "f = 10*sin(8*xx*(yy-1))\n",
    "D2 = dot(D,D); D2 = D2[1:N,1:N]; I = eye(N-1)\n",
    "L = kron(I,D2) + kron(D2,I)\n",
    "# Plot sparsity pattern\n",
    "figure(figsize=(8,8)), spy(L)\n",
    "# Solve Lu=f\n",
    "u = solve(L,f)\n",
    "# Convert 1-d vectors to 2-d\n",
    "uu = zeros((N+1,N+1)); uu[1:N,1:N] = reshape(u,(N-1,N-1))\n",
    "[xx,yy] = meshgrid(x,y)\n",
    "value = uu[N//4,N//4]\n",
    "\n",
    "# Interpolate to finer mesh just for visualization\n",
    "f = interp2d(x,y,uu,kind='cubic')\n",
    "xxx = linspace(-1.0,1.0,50)\n",
    "uuu = f(xxx,xxx)\n",
    "fig = figure(figsize=(8,8))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "X,Y = meshgrid(xxx,xxx)\n",
    "ax.plot_surface(X,Y,uuu,rstride=1,cstride=1,cmap=cm.jet,edgecolor='black')\n",
    "title(\"$u(2^{-1/2},2^{-1/2})$=\"+str(value))\n",
    "xlabel(\"x\"); ylabel(\"y\");"
   ]
  }
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