Source code for FIAT.bell
# Copyright (C) 2018 Robert C. Kirby
#
# This file is part of FIAT (https://www.fenicsproject.org)
#
# SPDX-License-Identifier: LGPL-3.0-or-later
# This is not quite Bell, but is 21-dofs and includes 3 extra constraint
# functionals. The first 18 basis functions are the reference element
# bfs, but the extra three are used in the transformation theory.
from FIAT import finite_element, polynomial_set, dual_set, functional
from FIAT.reference_element import TRIANGLE, ufc_simplex
[docs]class BellDualSet(dual_set.DualSet):
def __init__(self, ref_el):
entity_ids = {}
nodes = []
cur = 0
# make nodes by getting points
# need to do this dimension-by-dimension, facet-by-facet
top = ref_el.get_topology()
verts = ref_el.get_vertices()
sd = ref_el.get_spatial_dimension()
if ref_el.get_shape() != TRIANGLE:
raise ValueError("Bell only defined on triangles")
pd = functional.PointDerivative
# get jet at each vertex
entity_ids[0] = {}
for v in sorted(top[0]):
nodes.append(functional.PointEvaluation(ref_el, verts[v]))
# first derivatives
for i in range(sd):
alpha = [0] * sd
alpha[i] = 1
nodes.append(pd(ref_el, verts[v], alpha))
# second derivatives
alphas = [[2, 0], [1, 1], [0, 2]]
for alpha in alphas:
nodes.append(pd(ref_el, verts[v], alpha))
entity_ids[0][v] = list(range(cur, cur + 6))
cur += 6
# we need an edge quadrature rule for the moment
from FIAT.quadrature_schemes import create_quadrature
from FIAT.jacobi import eval_jacobi
rline = ufc_simplex(1)
q1d = create_quadrature(rline, 8)
q1dpts = q1d.get_points()
leg4_at_qpts = eval_jacobi(0, 0, 4, 2.0*q1dpts - 1)
imond = functional.IntegralMomentOfNormalDerivative
entity_ids[1] = {}
for e in sorted(top[1]):
entity_ids[1][e] = [18+e]
nodes.append(imond(ref_el, e, q1d, leg4_at_qpts))
entity_ids[2] = {0: []}
super(BellDualSet, self).__init__(nodes, ref_el, entity_ids)
[docs]class Bell(finite_element.CiarletElement):
"""The Bell finite element."""
def __init__(self, ref_el):
poly_set = polynomial_set.ONPolynomialSet(ref_el, 5)
dual = BellDualSet(ref_el)
super(Bell, self).__init__(poly_set, dual, 5)