p23: Eigenvalues of Laplacian in 2-D

We solve the following eigenvalue problem

-(u_{xx} + u_{yy}) = \lambda u, \qquad -1 < x,y < 1, \qquad u=0 \quad \mbox{on boundary}

(1)

#%config InlineBackend.figure_format='svg'
from chebPy import cheb
from numpy import meshgrid,dot,eye,kron,zeros,reshape,pi,real,imag
from numpy import diag,exp,argsort,linspace,inf
from matplotlib.pyplot import figure,subplot,plot,title,contour
from scipy.linalg import eig,norm
from scipy.interpolate import RegularGridInterpolator

# Set up tensor product Laplacian and compute 4 eigenmodes
N = 16; D,x = cheb(N); y = x;
xx,yy = meshgrid(x[1:N],y[1:N],indexing='ij')
xx = reshape(xx,(N-1)**2, order='F')
yy = reshape(yy,(N-1)**2, order='F')
D2 = dot(D,D); D2 = D2[1:N,1:N]; I = eye(N-1)
L = -kron(I,D2) - kron(D2,I)
D,V = eig(L); D = real(D); V = real(V)
ii = argsort(D); ii = ii[0:4]; D = D[ii]; V = V[:,ii]

# Reshape them to 2D grid, interpolate to finer grid, and plot
fine = linspace(-1.0,1.0,100,True);
X,Y = meshgrid(fine,fine,indexing='ij')
uu = zeros((N+1,N+1));

figure(figsize=(10,10))
for i in range(4):
    uu[1:N,1:N] = reshape(V[:,i],(N-1,N-1),order='F')
    uu = uu/norm(uu,inf)
    f = RegularGridInterpolator((x,y),uu,method='cubic')
    uuu = f((X,Y))
    subplot(2,2,i+1)
    contour(X,Y,uuu,10)
    title(r"eig = %18.12f $\pi^2/4$"%(D[i]/(pi**2/4)))