p16.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p16: Poisson equation in 2-D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We solve the following Poisson problem\n",
    "\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": [
    "%config InlineBackend.figure_format='svg'\n",
    "from chebPy import cheb\n",
    "from numpy import meshgrid,sin,dot,eye,kron,zeros,reshape,linspace\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 RegularGridInterpolator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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   "source": [
    "N = 24; D,x = cheb(N); y = x;\n",
    "xx,yy = meshgrid(x[1:N],y[1:N],indexing='ij')\n",
    "xx = reshape(xx,(N-1)**2,order='F')\n",
    "yy = reshape(yy,(N-1)**2,order='F')\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=(6,6)), 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),order='F')\n",
    "[xx,yy] = meshgrid(x,y,indexing='ij')\n",
    "value = uu[3*N//4,3*N//4]\n",
    "\n",
    "# Interpolate to finer mesh just for visualization\n",
    "f = RegularGridInterpolator((x,y),uu,method='cubic')\n",
    "xxx = linspace(-1.0,1.0,50); X,Y = meshgrid(xxx,xxx,indexing='ij')\n",
    "uuu = f((X,Y))\n",
    "fig = figure(figsize=(6,6))\n",
    "ax = fig.add_subplot(111, projection='3d')\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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