"""Sobolev spaces.
This module defines a symbolic heirarchy of Sobolev spaces to enable
symbolic reasoning about the spaces in which finite elements lie.
"""
# Copyright (C) 2014 Imperial College London and others
#
# This file is part of UFL (https://www.fenicsproject.org)
#
# SPDX-License-Identifier: LGPL-3.0-or-later
#
# Written by David Ham 2014
#
# Modified by Martin Alnaes 2014
# Modified by Lizao Li 2015
# Modified by Thomas Gibson 2017
from functools import total_ordering
from math import inf, isinf
__all_classes__ = ["SobolevSpace", "DirectionalSobolevSpace"]
[docs]@total_ordering
class SobolevSpace(object):
"""Symbolic representation of a Sobolev space.
This implements a subset of the methods of a Python set so that
finite elements and other Sobolev spaces can be tested for
inclusion.
"""
def __init__(self, name, parents=None):
"""Instantiate a SobolevSpace object.
Args:
name: The name of this space,
parents: A set of Sobolev spaces of which this
space is a subspace.
"""
self.name = name
p = frozenset(parents or [])
# Ensure that the inclusion operations are transitive.
self.parents = p.union(*[p_.parents for p_ in p])
self._order = {
"L2": 0,
"H1": 1,
"H2": 2,
"HInf": inf,
# Order for the elements below is taken from
# its parent Sobolev space
"HDiv": 0,
"HCurl": 0,
"HEin": 0,
"HDivDiv": 0,
"DirectionalH": 0,
}[self.name]
def __str__(self):
"""Format as a string."""
return self.name
def __repr__(self):
"""Representation."""
return f"SobolevSpace({self.name!r}, {list(self.parents)!r})"
def __eq__(self, other):
"""Check equality."""
return isinstance(other, SobolevSpace) and self.name == other.name
def __ne__(self, other):
"""Not equal."""
return not self == other
def __hash__(self):
"""Hash."""
return hash(("SobolevSpace", self.name))
def __getitem__(self, spatial_index):
"""Returns the Sobolev space associated with a particular spatial coordinate."""
return self
def __contains__(self, other):
"""Implement `fe in s` where `fe` is a FiniteElement and `s` is a SobolevSpace."""
if isinstance(other, SobolevSpace):
raise TypeError(
"Unable to test for inclusion of a SobolevSpace in another SobolevSpace. "
"Did you mean to use <= instead?"
)
return other.sobolev_space == self or self in other.sobolev_space.parents
def __lt__(self, other):
"""In common with intrinsic Python sets, < indicates "is a proper subset of"."""
return other in self.parents
[docs]@total_ordering
class DirectionalSobolevSpace(SobolevSpace):
"""Directional Sobolev space.
Symbolic representation of a Sobolev space with varying smoothness
in different spatial directions.
"""
def __init__(self, orders):
"""Instantiate a DirectionalSobolevSpace object.
Args:
orders: an iterable of orders of weak derivatives, where
the position denotes in what spatial variable the
smoothness requirement is enforced.
"""
assert all(
isinstance(x, int) or isinf(x) for x in orders
), "Order must be an integer or infinity."
name = "DirectionalH"
parents = [L2]
super(DirectionalSobolevSpace, self).__init__(name, parents)
self._orders = tuple(orders)
self._spatial_indices = range(len(self._orders))
def __getitem__(self, spatial_index):
"""Returns the Sobolev space associated with a particular spatial coordinate."""
if spatial_index not in range(len(self._orders)):
raise IndexError("Spatial index out of range.")
spaces = {0: L2, 1: H1, 2: H2, inf: HInf}
return spaces[self._orders[spatial_index]]
def __contains__(self, other):
"""Check if one space is contained in another.
Implement `fe in s` where `fe` is a FiniteElement and `s` is a
DirectionalSobolevSpace.
"""
if isinstance(other, SobolevSpace):
raise TypeError(
"Unable to test for inclusion of a SobolevSpace in another SobolevSpace. "
"Did you mean to use <= instead?"
)
return other.sobolev_space == self or all(
self[i] in other.sobolev_space.parents for i in self._spatial_indices
)
def __eq__(self, other):
"""Check equality."""
if isinstance(other, DirectionalSobolevSpace):
return self._orders == other._orders
return all(self[i] == other for i in self._spatial_indices)
def __lt__(self, other):
"""In common with intrinsic Python sets, < indicates "is a proper subset of."""
if isinstance(other, DirectionalSobolevSpace):
if self._spatial_indices != other._spatial_indices:
return False
return any(self._orders[i] > other._orders[i] for i in self._spatial_indices)
if other in [HDiv, HCurl]:
return all(self._orders[i] >= 1 for i in self._spatial_indices)
elif other.name in ["HDivDiv", "HEin"]:
# Don't know how these spaces compare
return NotImplementedError(f"Don't know how to compare with {other.name}")
else:
return any(self._orders[i] > other._order for i in self._spatial_indices)
def __str__(self):
"""Format as a string."""
return f"{self.name}({', '.join(map(str, self._orders))})"
L2 = SobolevSpace("L2")
HDiv = SobolevSpace("HDiv", [L2])
HCurl = SobolevSpace("HCurl", [L2])
H1 = SobolevSpace("H1", [HDiv, HCurl, L2])
H2 = SobolevSpace("H2", [H1, HDiv, HCurl, L2])
HInf = SobolevSpace("HInf", [H2, H1, HDiv, HCurl, L2])
HEin = SobolevSpace("HEin", [L2])
HDivDiv = SobolevSpace("HDivDiv", [L2])