L2_error_dual.py¶

Compute the $L^2$ -error between an infinite dimensional variable and its finite dimensional (discrete) projection in the mimetic polynomial space.

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class L2_error_dual.L2ErrorDual(bf, ct, quad_degree=None)[source]¶

A wrapper of all functions for computing $L^2$ -error. in the dual spaces $\widetilde{\text{NP}}_{N}(\Omega)$ , $\widetilde{\text{EP}}_{N-1}(\Omega)$ , $\widetilde{\text{FP}}_{N-1}(\Omega)$ and $\widetilde{\text{VP}}_{N-1}(\Omega)$ .

Parameters:

bf (MimeticBasisPolynomials) – The basis functions in $\Omega_{\mathrm{ref}}$ .
ct (CoordinateTransformation) –
The coordinate transformation representing the mapping $\Phi$ ,

$\Phi: \Omega_{\mathrm{ref}}\to\Omega$
quad_degree (list, tuple) – (default: None) The degree used for the numerical integral. It should be a list or tuple of three positive integers. If it is None, a suitable degree will be obtained from bf.

Example:

>>> import numpy as np
>>> from numpy import sin, cos, pi
>>> from coordinate_transformation import CoordinateTransformation
>>> from coordinate_transformation import Phi, d_Phi
>>> from mimetic_basis_polynomials import MimeticBasisPolynomials
>>> from projection_dual import ReductionDual
>>> ct = CoordinateTransformation(Phi, d_Phi)
>>> bf = MimeticBasisPolynomials('Lobatto-5','Lobatto-5','Lobatto-5')
>>> def pressure(x,y,z):
...     return sin(np.pi*x) * sin(pi*y) * sin(pi*z)
>>> def velocity_x(x,y,z):
...     return pi * cos(pi*x) * sin(pi*y) * sin(pi*z)
>>> def velocity_y(x,y,z):
...     return pi * sin(pi*x) * cos(pi*y) * sin(pi*z)
>>> def velocity_z(x,y,z):
...     return pi * sin(pi*x) * sin(pi*y) * cos(pi*z)
>>> def source(x,y,z):
...     return -3 * pi**2 * sin(np.pi*x) * sin(pi*y) * sin(pi*z)
>>> rd = ReductionDual(bf, ct)
>>> loc_dofs_N = rd.NP(pressure)
>>> loc_dofs_E = rd.EP((velocity_x, velocity_y, velocity_z))
>>> loc_dofs_F = rd.FP((velocity_x, velocity_y, velocity_z))
>>> loc_dofs_V = rd.VP(source)
>>> L2e = L2ErrorDual(bf, ct)
>>> L2e.NP(loc_dofs_N, pressure) 
0.0228947239...
>>> L2e.EP(loc_dofs_E, (velocity_x, velocity_y, velocity_z)) 
0.3612213119...
>>> L2e.FP(loc_dofs_F, (velocity_x, velocity_y, velocity_z)) 
0.4178056796...
>>> L2e.VP(loc_dofs_V, source) 
2.8203143473...

EP(loc_dofs, exact_solution)[source]¶

Let $\boldsymbol{\omega}\in H^1(\Omega)$ and $\boldsymbol{\omega}^h=\pi\left( \boldsymbol{\omega}\right)\in \widetilde{\text{EP}}_{N-1}(\Omega)$ , we compute $\left\|\boldsymbol{\omega}^h-\boldsymbol{\omega} \right\|_{L^2\text{-error}}$ .

Parameters:

loc_dofs – The local dofs of $\boldsymbol{\omega}^h$ .
exact_solution – A tuple of three functions representing the three components of the vector $\boldsymbol{\omega}$ .

Returns:

A float representing the $L^2$ -error.

FP(loc_dofs, exact_solution)[source]¶

Let $\boldsymbol{\omega}\in H^1(\Omega)$ and $\boldsymbol{u}^h=\pi\left( \boldsymbol{u}\right)\in \widetilde{\text{FP}}_{N-1}(u)$ , we compute $\left\|\boldsymbol{u}^h-\boldsymbol{u} \right\|_{L^2\text{-error}}$ .

Parameters:

loc_dofs – The local dofs of $\boldsymbol{u}^h$ .
exact_solution – A tuple of three functions representing the three components of the vector $\boldsymbol{u}$ .

Returns:

A float representing the $L^2$ -error.

NP(loc_dofs, exact_solution)[source]¶

Let $\psi\in H^1(\Omega)$ and $\psi^h=\pi\left(\psi\right)\in \widetilde{\text{NP}}_{N}(\Omega)$ , we compute $\left\|\psi^h-\psi\right\|_{L^2\text{-error}}$ .