# Copyright (C) 2008-2012 Robert C. Kirby (Texas Tech University)
# Modified by Andrew T. T. McRae (Imperial College London)
#
# This file is part of FIAT (https://www.fenicsproject.org)
#
# SPDX-License-Identifier: LGPL-3.0-or-later
from FIAT import (expansions, polynomial_set, quadrature, dual_set,
finite_element, functional)
import numpy
from itertools import chain
from FIAT.check_format_variant import check_format_variant
[docs]def RTSpace(ref_el, deg):
"""Constructs a basis for the the Raviart-Thomas space
(P_k)^d + P_k x"""
sd = ref_el.get_spatial_dimension()
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, deg + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, deg + 1)
dimPk = expansions.polynomial_dimension(ref_el, deg)
dimPkm1 = expansions.polynomial_dimension(ref_el, deg - 1)
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk_from_Pkp1 = vec_Pkp1.take(vec_Pk_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, deg + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * deg + 2)
# have to work on this through "tabulate" interface
# first, tabulate PkH at quadrature points
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkH_at_Qpts = PkH.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
PkHx_coeffs = numpy.zeros((PkH.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
for i in range(PkH.get_num_members()):
for j in range(sd):
fooij = PkH_at_Qpts[i, :] * Qpts[:, j] * Qwts
PkHx_coeffs[i, j, :] = numpy.dot(Pkp1_at_Qpts, fooij)
PkHx = polynomial_set.PolynomialSet(ref_el,
deg,
deg + 1,
vec_Pkp1.get_expansion_set(),
PkHx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk_from_Pkp1, PkHx)
[docs]class RTDualSet(dual_set.DualSet):
"""Dual basis for Raviart-Thomas elements consisting of point
evaluation of normals on facets of codimension 1 and internal
moments against polynomials"""
def __init__(self, ref_el, degree, variant, quad_deg):
entity_ids = {}
nodes = []
sd = ref_el.get_spatial_dimension()
t = ref_el.get_topology()
if variant == "integral":
facet = ref_el.get_facet_element()
# Facet nodes are \int_F v\cdot n p ds where p \in P_{q-1}
# degree is q - 1
Q = quadrature.make_quadrature(facet, quad_deg)
Pq = polynomial_set.ONPolynomialSet(facet, degree)
Pq_at_qpts = Pq.tabulate(Q.get_points())[tuple([0]*(sd - 1))]
for f in range(len(t[sd - 1])):
for i in range(Pq_at_qpts.shape[0]):
phi = Pq_at_qpts[i, :]
nodes.append(functional.IntegralMomentOfScaledNormalEvaluation(ref_el, Q, phi, f))
# internal nodes. These are \int_T v \cdot p dx where p \in P_{q-2}^d
if degree > 0:
Q = quadrature.make_quadrature(ref_el, quad_deg)
qpts = Q.get_points()
Pkm1 = polynomial_set.ONPolynomialSet(ref_el, degree - 1)
zero_index = tuple([0 for i in range(sd)])
Pkm1_at_qpts = Pkm1.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm1_at_qpts.shape[0]):
phi_cur = Pkm1_at_qpts[i, :]
l_cur = functional.IntegralMoment(ref_el, Q, phi_cur, (d,), (sd,))
nodes.append(l_cur)
elif variant == "point":
# codimension 1 facets
for i in range(len(t[sd - 1])):
pts_cur = ref_el.make_points(sd - 1, i, sd + degree)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointScaledNormalEvaluation(ref_el, i, pt_cur)
nodes.append(f)
# internal nodes. Let's just use points at a lattice
if degree > 0:
cpe = functional.ComponentPointEvaluation
pts = ref_el.make_points(sd, 0, degree + sd)
for d in range(sd):
for i in range(len(pts)):
l_cur = cpe(ref_el, d, (sd,), pts[i])
nodes.append(l_cur)
# sets vertices (and in 3d, edges) to have no nodes
for i in range(sd - 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# set codimension 1 (edges 2d, faces 3d) dof
pts_facet_0 = ref_el.make_points(sd - 1, 0, sd + degree)
pts_per_facet = len(pts_facet_0)
entity_ids[sd - 1] = {}
for i in range(len(t[sd - 1])):
entity_ids[sd - 1][i] = list(range(cur, cur + pts_per_facet))
cur += pts_per_facet
# internal nodes, if applicable
entity_ids[sd] = {0: []}
if degree > 0:
num_internal_nodes = expansions.polynomial_dimension(ref_el,
degree - 1)
entity_ids[sd][0] = list(range(cur, cur + num_internal_nodes * sd))
super(RTDualSet, self).__init__(nodes, ref_el, entity_ids)
[docs]class RaviartThomas(finite_element.CiarletElement):
"""
The Raviart Thomas element
:arg ref_el: The reference element.
:arg k: The degree.
:arg variant: optional variant specifying the types of nodes.
variant can be chosen from ["point", "integral", "integral(quadrature_degree)"]
"point" -> dofs are evaluated by point evaluation. Note that this variant has suboptimal
convergence order in the H(div)-norm
"integral" -> dofs are evaluated by quadrature rule. The quadrature degree is chosen to integrate
polynomials of degree 5*k so that most expressions will be interpolated exactly. This is important
when you want to have (nearly) divergence-preserving interpolation.
"integral(quadrature_degree)" -> dofs are evaluated by quadrature rule of degree quadrature_degree
"""
def __init__(self, ref_el, k, variant=None):
degree = k - 1
(variant, quad_deg) = check_format_variant(variant, degree, "Raviart Thomas")
poly_set = RTSpace(ref_el, degree)
dual = RTDualSet(ref_el, degree, variant, quad_deg)
formdegree = ref_el.get_spatial_dimension() - 1 # (n-1)-form
super(RaviartThomas, self).__init__(poly_set, dual, degree, formdegree,
mapping="contravariant piola")