Mimetic Spectral Elements

FIAT/__init__.py

@@ -39,6 +39,8 @@
 from FIAT.mixed import MixedElement # noqa: F401
 from FIAT.restricted import RestrictedElement # noqa: F401
 from FIAT.quadrature_element import QuadratureElement # noqa: F401
+from FIAT.extended_gauss_legendre import ExtendedGaussLegendre
+from FIAT.edge import EdgeGaussLobattoLegendre, EdgeExtendedGaussLegendre
 
 # Important functionality
 from FIAT.quadrature import make_quadrature # noqa: F401
@@ -66,6 +68,9 @@
  "Lagrange": Lagrange,
  "Gauss-Lobatto-Legendre": GaussLobattoLegendre,
  "Gauss-Legendre": GaussLegendre,
+ "Extended-Gauss-Legendre": ExtendedGaussLegendre,
+ "Gauss-Lobatto-Legendre Edge": EdgeGaussLobattoLegendre,
+ "Extended-Gauss-Legendre Edge": EdgeExtendedGaussLegendre,
  "Gauss-Radau": GaussRadau,
  "Morley": Morley,
  "Nedelec 1st kind H(curl)": Nedelec,

FIAT/edge.py

@@ -0,0 +1,176 @@
+# Copyright (C) 2019 Chris Eldred (Inria Grenoble Rhone-Alpes) and Werner Bauer (Inria Rennes)
+#
+# This file is part of FIAT.
+#
+# FIAT 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.
+#
+# FIAT 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 FIAT. If not, see <http://www.gnu.org/licenses/>.
+#
+
+import sympy
+import numpy as np
+from FIAT.finite_element import FiniteElement
+from FIAT.dual_set import make_entity_closure_ids
+from FIAT.polynomial_set import mis
+from FIAT import quadrature
+
+x = sympy.symbols('x')
+
+
+def _lagrange_poly(i, xi):
+ '''Returns the Langrange polynomial P_i(x) of pts xi
+ which interpolates the values (0,0,..1,..,0) where 1 is at the ith support point
+ :param i: non-zero location
+ :param xi: set of N support points
+ '''
+ index = list(range(len(xi)))
+ index.pop(i)
+ return sympy.prod([(x-xi[j][0])/(xi[i][0]-xi[j][0]) for j in index])
+
+
+def _lagrange_basis(spts):
+ symbas = []
+ for i in range(len(spts)):
+ symbas.append(_lagrange_poly(i, spts))
+ return symbas
+
+
+def _create_compatible_l2_basis(cg_symbas):
+ ncg_basis = len(cg_symbas)
+ symbas = []
+ for i in range(1, ncg_basis):
+ basis = 0
+ for j in range(i):
+ basis = basis + sympy.diff(-cg_symbas[j])
+ symbas.append(basis)
+ return symbas
+
+
+class _EdgeElement(FiniteElement):
+
+ def __new__(cls, ref_el, degree):
+ dim = ref_el.get_spatial_dimension()
+ if dim == 1:
+ self = super().__new__(cls)
+ return self
+ else:
+ raise IndexError("only intervals supported for _IntervalElement")
+
+ def tabulate(self, order, points, entity=None):
+
+ if entity is None:
+ entity = (self.ref_el.get_spatial_dimension(), 0)
+ entity_dim, entity_id = entity
+ dim = self.ref_el.get_spatial_dimension()
+
+ transform = self.ref_el.get_entity_transform(entity_dim, entity_id)
+ points = list(map(transform, points))
+
+ phivals = {}
+ for o in range(order + 1):
+ alphas = mis(dim, o)
+ for alpha in alphas:
+ try:
+ basis = self.basis[alpha]
+ except KeyError:
+ basis = sympy.diff(self.basis[(0,)], x)
+ self.basis[alpha] = basis
+ T = np.zeros((len(basis), len(points)))
+ for i in range(len(points)):
+ subs = {x: points[i][0]}
+ for j, f in enumerate(basis):
+ T[j, i] = f.evalf(subs=subs)
+ phivals[alpha] = T
+
+ return phivals
+
+ def degree(self):
+ return self._degree
+
+ def get_nodal_basis(self):
+ raise NotImplementedError("get_nodal_basis not implemented for _IntervalElement")
+
+ def get_dual_set(self):
+ raise NotImplementedError("get_dual_set is not implemented for _IntervalElement")
+
+ def get_coeffs(self):
+ raise NotImplementedError("get_coeffs not implemented for _IntervalElement")
+
+ def entity_dofs(self):
+ """Return the map of topological entities to degrees of
+ freedom for the finite element."""
+ return self.entity_ids
+
+ def entity_closure_dofs(self):
+ """Return the map of topological entities to degrees of
+ freedom on the closure of those entities for the finite element."""
+ return self.entity_closure_ids
+
+ def value_shape(self):
+ return ()
+
+ def dmats(self):
+ raise NotImplementedError
+
+ def get_num_members(self, arg):
+ raise NotImplementedError
+
+ def space_dimension(self):
+ return len(self.basis[(0,)])
+
+ def __init__(self, ref_el, degree):
+
+ dim = ref_el.get_spatial_dimension()
+ topology = ref_el.get_topology()
+
+ if not dim == 1:
+ raise IndexError("only intervals supported for DMSE")
+
+ formdegree = 1
+
+ # This is general code to build empty entity_ids
+ entity_ids = {}
+ for dim in range(dim+1):
+ entity_ids[dim] = {}
+ for entity in sorted(topology[dim]):
+ entity_ids[dim][entity] = []
+
+ # The only dofs for DMSE are with the interval!
+ entity_ids[dim][0] = list(range(degree + 1))
+
+ # Build the basis
+ # This is a dictionary basis[o] that gives the "basis" functions for derivative order o, where o=0 is just the basis itself
+ # This is filled as needed for o > 0 by tabulate
+
+ self.entity_ids = entity_ids
+ self.entity_closure_ids = make_entity_closure_ids(ref_el, entity_ids)
+ self._degree = degree
+
+ super(_EdgeElement, self).__init__(ref_el=ref_el, dual=None, order=degree, formdegree=formdegree)
+
+
+class EdgeGaussLobattoLegendre(_EdgeElement):
+ def __init__(self, ref_el, degree):
+ super(EdgeGaussLobattoLegendre, self).__init__(ref_el, degree)
+ cont_pts = quadrature.GaussLobattoLegendreQuadratureLineRule(ref_el, degree+2).pts
+ cont_basis = _lagrange_basis(cont_pts)
+ basis = _create_compatible_l2_basis(cont_basis)
+ self.basis = {(0,): sympy.Array(basis)}
+
+
+class EdgeExtendedGaussLegendre(_EdgeElement):
+ def __init__(self, ref_el, degree):
+ super(EdgeExtendedGaussLegendre, self).__init__(ref_el, degree)
+ cont_pts = quadrature.ExtendedGaussLegendreQuadratureLineRule(ref_el, degree+2).pts
+ cont_basis = _lagrange_basis(cont_pts)
+ basis = _create_compatible_l2_basis(cont_basis)
+ self.basis = {(0,): sympy.Array(basis)}

FIAT/extended_gauss_legendre.py

@@ -0,0 +1,44 @@
+# Copyright (C) 2019 Chris Eldred (Inria Grenoble Rhone-Alpes) and Werner Bauer (Inria Rennes)
+#
+# This file is part of FIAT.
+#
+# FIAT 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.
+#
+# FIAT 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 FIAT. If not, see <http://www.gnu.org/licenses/>.
+#
+# Written by Chris Eldred (Inria Grenoble Rhone-Alpes) and Werner Bauer (Inria Rennes) 2019
+
+from FIAT import finite_element, polynomial_set, dual_set, functional, quadrature
+from FIAT.reference_element import LINE
+
+
+class ExtendedGaussLegendreDualSet(dual_set.DualSet):
+ """The dual basis for 1D continuous elements with nodes at the
+ Extended-Gauss-Legendre points."""
+ def __init__(self, ref_el, degree):
+ entity_ids = {0: {0: [0], 1: [degree]},
+ 1: {0: list(range(1, degree))}}
+ lr = quadrature.ExtendedGaussLegendreQuadratureLineRule(ref_el, degree+1)
+ nodes = [functional.PointEvaluation(ref_el, x) for x in lr.pts]
+
+ super(ExtendedGaussLegendreDualSet, self).__init__(nodes, ref_el, entity_ids)
+
+
+class ExtendedGaussLegendre(finite_element.CiarletElement):
+ """1D continuous element with nodes at the Extended-Gauss-Legendre points."""
+ def __init__(self, ref_el, degree):
+ if ref_el.shape != LINE:
+ raise ValueError("Extended-Gauss-Legendre elements are only defined in one dimension.")
+ poly_set = polynomial_set.ONPolynomialSet(ref_el, degree)
+ dual = ExtendedGaussLegendreDualSet(ref_el, degree)
+ formdegree = 0 # 0-form
+ super(ExtendedGaussLegendre, self).__init__(poly_set, dual, degree, formdegree)

FIAT/quadrature.py

@@ -97,7 +97,7 @@ def __init__(self, ref_el, m):
 
 
 class GaussLegendreQuadratureLineRule(QuadratureRule):
- """Produce the Gauss--Legendre quadrature rules on the interval using
+ """Produce the Gauss-Legendre quadrature rules on the interval using
  the implementation in numpy. This facilitates implementing
  discontinuous spectral elements.
 
@@ -123,6 +123,39 @@ def __init__(self, ref_el, m):
  QuadratureRule.__init__(self, ref_el, xs, ws)
 
 
+class ExtendedGaussLegendreQuadratureLineRule(QuadratureRule):
+ """Produce the Extended-Gauss-Legendre quadrature rules on the interval using
+ the implementation in numpy. This facilitates implementing
+ dual continuous mimetic spectral elements.
+
+ The quadrature rule uses m points for a degree of precision of 2(m-2)-3 = 2m - 7.
+ Thus, not recommend for actual use (but is useful to define the Extended-Gauss-Legendre elements)
+ """
+ def __init__(self, ref_el, m):
+ if m < 3:
+ raise ValueError(
+ "Extended-Gauss-Legendre quadrature invalid for fewer than 3 points")
+
+ xs_ref, ws_ref = numpy.polynomial.legendre.leggauss(m-2)
+ A, b = reference_element.make_affine_mapping(((-1.,), (1.)),
+ ref_el.get_vertices())
+
+ mapping = lambda x: numpy.dot(A, x) + b
+
+ scale = numpy.linalg.det(A)
+
+ ws = list([scale * w for w in ws_ref])
+ xs = [tuple(mapping(x_ref)[0]) for x_ref in xs_ref]
+
+ xs.insert(0, (0.,))
+ xs.append((1.,))
+ # The integral of these basis functions over the interval is 0!
+ ws.insert(0, 0.)
+ ws.append(0.)
+
+ QuadratureRule.__init__(self, ref_el, tuple(xs), tuple(ws))
+
+
 class RadauQuadratureLineRule(QuadratureRule):
  """Produce the Gauss--Radau quadrature rules on the interval using
  an adaptation of Winkel's Matlab code.

test/unit/test_edge.py

@@ -0,0 +1,72 @@
+# Copyright (C) 2016 Imperial College London and others
+#
+# This file is part of FIAT.
+#
+# FIAT 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.
+#
+# FIAT 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 FIAT. If not, see <http://www.gnu.org/licenses/>.
+#
+# Authors:
+#
+# Christopher Eldred
+
+import pytest
+import math
+import sympy
+import FIAT
+
+x = sympy.symbols('x')
+
+
+@pytest.mark.parametrize("degree", list(range(0, 7)))
+def test_gll_edge_basis_values(degree):
+ """Ensure that edge elements are histopolatory"""
+
+ s = FIAT.ufc_simplex(1)
+
+ fe = FIAT.EdgeGaussLobattoLegendre(s, degree)
+
+ basis = fe.basis[(0,)]
+ cont_pts = FIAT.quadrature.GaussLobattoLegendreQuadratureLineRule(s, degree + 2).pts
+
+ for i in range(len(cont_pts)-1):
+ for j in range(basis.shape[0]):
+ int_sub = sympy.integrate(basis[j], (x, cont_pts[i][0], cont_pts[i+1][0]))
+ if i == j:
+ assert(math.isclose(int_sub, 1.))
+ else:
+ assert(math.isclose(int_sub, 0., abs_tol=1e-9))
+
+
+@pytest.mark.parametrize("degree", list(range(2, 7)))
+def test_egl_edge_basis_values(degree):
+ """Ensure that edge elements are histopolatory"""
+
+ s = FIAT.ufc_simplex(1)
+
+ fe = FIAT.EdgeExtendedGaussLegendre(s, degree)
+
+ basis = fe.basis[(0,)]
+ cont_pts = FIAT.quadrature.ExtendedGaussLegendreQuadratureLineRule(s, degree + 2).pts
+
+ for i in range(len(cont_pts)-1):
+ for j in range(basis.shape[0]):
+ int_sub = sympy.integrate(basis[j], (x, cont_pts[i][0], cont_pts[i+1][0]))
+ if i == j:
+ assert(math.isclose(int_sub, 1.))
+ else:
+ assert(math.isclose(int_sub, 0., abs_tol=1e-9))
+
+
+if __name__ == '__main__':
+ import os
+ pytest.main(os.path.abspath(__file__))