p23.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p23: Eigenvalues of perturbed Laplacian on [-1,1]x[-1,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We solve the following eigenvalue problem\n",
    "$$\n",
    "-(u_{xx} + u_{yy}) + f(x,y) u = \\lambda u, \\qquad -1 < x,y < 1, \\qquad u=0 \\quad \\mbox{on boundary}\n",
    "$$\n",
    "where\n",
    "$$\n",
    "f(x,y) = \\exp(20(y-x-1))\n",
    "$$"
   ]
  },
  {
   "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,dot,eye,kron,zeros,reshape,pi,real,imag\n",
    "from numpy import diag,exp,argsort,linspace,inf\n",
    "from matplotlib.pyplot import figure,subplot,plot,title,contour\n",
    "from scipy.linalg import eig,norm\n",
    "from scipy.interpolate import interp2d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 720x720 with 4 Axes>"
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   "source": [
    "# Set up tensor product Laplacian and compute 4 eigenmodes\n",
    "N = 16; 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",
    "D2 = dot(D,D); D2 = D2[1:N,1:N]; I = eye(N-1)\n",
    "L = -kron(I,D2) - kron(D2,I)\n",
    "L = L + diag(exp(20*(yy-xx-1)))\n",
    "D,V = eig(L); D = real(D); V = real(V)\n",
    "ii = argsort(D); ii = ii[0:4]; D = D[ii]; V = V[:,ii]\n",
    "\n",
    "# Reshape them to 2D grid, interpolate to finer grid, and plot\n",
    "fine = linspace(-1.0,1.0,100,True);\n",
    "uu = zeros((N+1,N+1));\n",
    "\n",
    "figure(figsize=(10,10))\n",
    "for i in range(4):\n",
    "    uu[1:N,1:N] = reshape(V[:,i],(N-1,N-1))\n",
    "    uu = uu/norm(uu,inf)\n",
    "    f = interp2d(x,y,uu,kind='cubic')\n",
    "    uuu = f(fine,fine)\n",
    "    subplot(2,2,i+1)\n",
    "    contour(fine,fine,uuu,10)\n",
    "    title(\"eig = %18.12f $\\pi^2/4$\"%(D[i]/(pi**2/4)))\n",
    "    "
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