Composite Gauss Interpolation Error

Copyright (C) 2020 Andreas Kloeckner

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In [8]:
from __future__ import division
import numpy as np
import scipy as sp
import scipy.special as ss
import matplotlib.pyplot as pt
import numpy.linalg as la

Populating the interactive namespace from numpy and matplotlib

WARNING: pylab import has clobbered these variables: ['f']
`%pylab --no-import-all` prevents importing * from pylab and numpy
In [21]:
nelements = 5
nnodes = 3
mesh = np.linspace(-1, 1, nelements+1, endpoint=True)
gauss_nodes = ss.legendre(nnodes).weights[:, 0]*0.5 + 0.5

widths = np.diff(mesh)
nodes = mesh[:-1, np.newaxis] + widths[:, np.newaxis] * gauss_nodes
In [22]:
def f(x):
    return np.abs(x-0.123812378)

pt.plot(nodes.flat, f(nodes).flat)
Out[22]:
[<matplotlib.lines.Line2D at 0x107b3dc10>]
In [23]:
nmany_nodes = 32

many_gauss_nodes = ss.legendre(nmany_nodes).weights[:, 0]*0.5 + 0.5
many_nodes = mesh[:-1, np.newaxis] + widths[:, np.newaxis] * many_gauss_nodes

def legendre_vdm(nodes, nmodes):
    result = np.empty((len(nodes), nmodes))
    for i in xrange(nmodes):
        result[:, i] = ss.eval_legendre(i, nodes)
    return result

vdm = legendre_vdm(gauss_nodes, nnodes)
many_vdm = legendre_vdm(many_gauss_nodes, nnodes)
zero_pad = np.zeros((nmany_nodes, nnodes))
zero_pad[:nnodes, :nnodes] = np.eye(nnodes)
upterpolate = np.dot(many_vdm, la.inv(vdm))
In [24]:
fnodes = f(nodes)
fmany_nodes = np.dot(upterpolate, fnodes.T).T
pt.plot(many_nodes.flat, fmany_nodes.flat)
Out[24]:
[<matplotlib.lines.Line2D at 0x107bcc410>]
In [ ]: