Composite Gauss Interpolation Error

Copyright (C) 2020 Andreas Kloeckner

MIT License

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

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

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

def f(x):
    return np.abs(x-0.123812378)

pt.plot(nodes.flat, f(nodes).flat)

[<matplotlib.lines.Line2D at 0x107b3dc10>]

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))

fnodes = f(nodes)
fmany_nodes = np.dot(upterpolate, fnodes.T).T
pt.plot(many_nodes.flat, fmany_nodes.flat)

[<matplotlib.lines.Line2D at 0x107bcc410>]