/usr/lib/python2.7/dist-packages/dolfin/fem/norms.py is in python-dolfin 2016.2.0-2.
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"""This module provides a simple way to compute various norms of
:py:class:`Vectors <dolfin.cpp.Vector>` and :py:class:`Functions
<dolfin.functions.function.Function>`, including the standard
:math:`L^2`-norm and other norms."""
# Copyright (C) 2008-2014 Anders Logg
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see <http://www.gnu.org/licenses/>.
from six import string_types
import ufl
from ufl import inner, grad, div, curl, dx, FiniteElement, VectorElement, Coefficient
from math import sqrt
import dolfin.cpp as cpp
from dolfin.cpp import GenericVector, GenericFunction, Function, Mesh, error, Vector
from dolfin.fem.assembling import assemble
from dolfin.fem.interpolation import interpolate
from dolfin.functions.function import Function
from dolfin.functions.functionspace import FunctionSpace, \
VectorFunctionSpace, TensorFunctionSpace
__all__ = ["norm", "errornorm"]
def norm(v, norm_type="L2", mesh=None):
"""
Return the norm of a given vector or function.
*Arguments*
v
a :py:class:`Vector <dolfin.cpp.Vector>` or
a :py:class:`Function <dolfin.functions.function.Function>`.
norm_type
see below for alternatives.
mesh
optional :py:class:`Mesh <dolfin.cpp.Mesh>` on
which to compute the norm.
If the norm type is not specified, the standard :math:`L^2` -norm
is computed. Possible norm types include:
*Vectors*
================ ================= ================
Norm Usage
================ ================= ================
:math:`l^2` norm(x, 'l2') Default
:math:`l^1` norm(x, 'l1')
:math:`l^\infty` norm(x, 'linf')
================ ================= ================
*Functions*
================ ================= =================================
Norm Usage Includes the :math:`L^2` -term
================ ================= =================================
:math:`L^2` norm(v, 'L2') Yes
:math:`H^1` norm(v, 'H1') Yes
:math:`H^1_0` norm(v, 'H10') No
:math:`H` (div) norm(v, 'Hdiv') Yes
:math:`H` (div) norm(v, 'Hdiv0') No
:math:`H` (curl) norm(v, 'Hcurl') Yes
:math:`H` (curl) norm(v, 'Hcurl0') No
================ ================= =================================
*Examples of usage*
.. code-block:: python
v = Function(V)
x = v.vector()
print norm(x, 'linf') # print the infinity norm of vector x
n = norm(v) # compute L^2 norm of v
print norm(v, 'Hdiv') # print H(div) norm of v
n = norm(v, 'H1', mesh) # compute H^1 norm of v on given mesh
"""
if not isinstance(v, (GenericVector, GenericFunction)):
cpp.dolfin_error("norms.py",
"compute norm",
"expected a GenericVector or GenericFunction")
# Check arguments
if not isinstance(norm_type, string_types):
cpp.dolfin_error("norms.py",
"compute norm",
"Norm type must be a string, not " +
str(type(norm_type)))
if mesh is not None and not isinstance(mesh, cpp.Mesh):
cpp.dolfin_error("norms.py",
"compute norm",
"Expecting a Mesh, not " + str(type(mesh)))
# Get mesh from function
if isinstance(v, Function) and mesh is None:
mesh = v.function_space().mesh()
# Define integration measure and domain
dx = ufl.dx(mesh)
# Select norm type
if isinstance(v, GenericVector):
return v.norm(norm_type.lower())
elif (isinstance(v, Coefficient) and isinstance(v, GenericFunction)):
if norm_type.lower() == "l2":
M = v**2*dx
elif norm_type.lower() == "h1":
M = (v**2 + grad(v)**2)*dx
elif norm_type.lower() == "h10":
M = grad(v)**2*dx
elif norm_type.lower() == "hdiv":
M = (v**2 + div(v)**2)*dx
elif norm_type.lower() == "hdiv0":
M = div(v)**2*dx
elif norm_type.lower() == "hcurl":
M = (v**2 + curl(v)**2)*dx
elif norm_type.lower() == "hcurl0":
M = curl(v)**2*dx
else:
cpp.dolfin_error("norms.py",
"compute norm",
"Unknown norm type (\"%s\") for functions"
% str(norm_type))
else:
cpp.dolfin_error("norms.py",
"compute norm",
"Unknown object type. Must be a vector or a function")
# Assemble value
r = assemble(M, form_compiler_parameters={"representation": "quadrature"})
# Check value
if r < 0.0:
cpp.dolfin_error("norms.py",
"compute norm",
"Square of norm is negative, might be a round-off error")
elif r == 0.0:
return 0.0
else:
return sqrt(r)
def errornorm(u, uh, norm_type="l2", degree_rise=3, mesh=None):
"""
Compute and return the error :math:`e = u - u_h` in the given norm.
*Arguments*
u, uh
:py:class:`Functions <dolfin.functions.function.Function>`
norm_type
Type of norm. The :math:`L^2` -norm is default.
For other norms, see :py:func:`norm <dolfin.fem.norms.norm>`.
degree_rise
The number of degrees above that of u_h used in the
interpolation; i.e. the degree of piecewise polynomials used
to approximate :math:`u` and :math:`u_h` will be the degree
of :math:`u_h` + degree_raise.
mesh
Optional :py:class:`Mesh <dolfin.cpp.Mesh>` on
which to compute the error norm.
In simple cases, one may just define
.. code-block:: python
e = u - uh
and evalute for example the square of the error in the :math:`L^2` -norm by
.. code-block:: python
assemble(e**2*dx(mesh))
However, this is not stable w.r.t. round-off errors considering that
the form compiler may expand(#) the expression above to::
e**2*dx = u**2*dx - 2*u*uh*dx + uh**2*dx
and this might get further expanded into thousands of terms for
higher order elements. Thus, the error will be evaluated by adding
a large number of terms which should sum up to something close to
zero (if the error is small).
This module computes the error by first interpolating both
:math:`u` and :math:`u_h` to a common space (of high accuracy),
then subtracting the two fields (which is easy since they are
expressed in the same basis) and then evaluating the integral.
(#) If using the tensor representation optimizations.
The quadrature represenation does not suffer from this problem.
"""
# Check argument
if not isinstance(u, cpp.GenericFunction):
cpp.dolfin_error("norms.py",
"compute error norm",
"Expecting a Function or Expression for u")
if not isinstance(uh, cpp.Function):
cpp.dolfin_error("norms.py",
"compute error norm",
"Expecting a Function for uh")
# Get mesh
if isinstance(u, Function) and mesh is None:
mesh = u.function_space().mesh()
if isinstance(uh, Function) and mesh is None:
mesh = uh.function_space().mesh()
if mesh is None:
cpp.dolfin_error("norms.py",
"compute error norm",
"Missing mesh")
# Get rank
if not u.ufl_shape == uh.ufl_shape:
cpp.dolfin_error("norms.py",
"compute error norm",
"Value shapes don't match")
shape = u.ufl_shape
rank = len(shape)
# Check that uh is associated with a finite element
if uh.ufl_element().degree() is None:
cpp.dolfin_error("norms.py",
"compute error norm",
"Function uh must have a finite element")
# Degree for interpolation space. Raise degree with respect to uh.
degree = uh.ufl_element().degree() + degree_rise
# Check degree of 'exact' solution u
degree_u = u.ufl_element().degree()
if degree_u is not None and degree_u < degree:
cpp.warning("Degree of exact solution may be inadequate for accurate result in errornorm.")
# Create function space
if rank == 0:
V = FunctionSpace(mesh, "Discontinuous Lagrange", degree)
elif rank == 1:
V = VectorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
dim=shape[0])
elif rank > 1:
V = TensorFunctionSpace(mesh, "Discontinuous Lagrange", degree,
shape=shape)
# Interpolate functions into finite element space
pi_u = interpolate(u, V)
pi_uh = interpolate(uh, V)
# Compute the difference
e = Function(V)
e.assign(pi_u)
e.vector().axpy(-1.0, pi_uh.vector())
# Compute norm
return norm(e, norm_type=norm_type, mesh=mesh)
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