Integral variant for Regge/HHJ (#96)

* docstring

* cleanup test_fiat.py

* Regge integral dofs

* HHJ integral dofs in 2d/3d

* support variant="integral(q)"

* BidirectionalMoment dofs

* address review comments

* Cleanup H(div, S) elements (#98)

* implementation of Hu-Zhang element

---------

Co-authored-by: Francis Aznaran <faznaran@nd.edu>

Author: pbrubeck

FIAT/__init__.py

@@ -2,10 +2,17 @@
 evaluating arbitrary order Lagrange and many other elements.
 Simplices in one, two, and three dimensions are supported."""
 
-import pkg_resources
+# Important functionality
+from FIAT.reference_element import ufc_cell, ufc_simplex       # noqa: F401
+from FIAT.quadrature import make_quadrature                    # noqa: F401
+from FIAT.quadrature_schemes import create_quadrature          # noqa: F401
+from FIAT.finite_element import FiniteElement, CiarletElement  # noqa: F401
+from FIAT.hdivcurl import Hdiv, Hcurl                          # noqa: F401
+from FIAT.mixed import MixedElement                            # noqa: F401
+from FIAT.restricted import RestrictedElement                  # noqa: F401
+from FIAT.quadrature_element import QuadratureElement          # noqa: F401
 
 # Import finite element classes
-from FIAT.finite_element import FiniteElement, CiarletElement  # noqa: F401
 from FIAT.argyris import Argyris
 from FIAT.bernardi_raugel import BernardiRaugel
 from FIAT.bernstein import Bernstein
@@ -17,8 +24,7 @@
 from FIAT.christiansen_hu import ChristiansenHu
 from FIAT.johnson_mercier import JohnsonMercier
 from FIAT.brezzi_douglas_marini import BrezziDouglasMarini
-from FIAT.Sminus import TrimmedSerendipityEdge  # noqa: F401
-from FIAT.Sminus import TrimmedSerendipityFace  # noqa: F401
+from FIAT.Sminus import TrimmedSerendipityEdge, TrimmedSerendipityFace  # noqa: F401
 from FIAT.SminusDiv import TrimmedSerendipityDiv  # noqa: F401
 from FIAT.SminusCurl import TrimmedSerendipityCurl  # noqa: F401
 from FIAT.brezzi_douglas_fortin_marini import BrezziDouglasFortinMarini
@@ -42,30 +48,22 @@
 from FIAT.raviart_thomas import RaviartThomas
 from FIAT.crouzeix_raviart import CrouzeixRaviart
 from FIAT.regge import Regge
+from FIAT.gopalakrishnan_lederer_schoberl import GopalakrishnanLedererSchoberlFirstKind
+from FIAT.gopalakrishnan_lederer_schoberl import GopalakrishnanLedererSchoberlSecondKind
 from FIAT.hellan_herrmann_johnson import HellanHerrmannJohnson
 from FIAT.arnold_winther import ArnoldWinther
 from FIAT.arnold_winther import ArnoldWintherNC
+from FIAT.hu_zhang import HuZhang
 from FIAT.mardal_tai_winther import MardalTaiWinther
 from FIAT.bubble import Bubble, FacetBubble
 from FIAT.tensor_product import TensorProductElement
 from FIAT.enriched import EnrichedElement
 from FIAT.nodal_enriched import NodalEnrichedElement
 from FIAT.discontinuous import DiscontinuousElement
 from FIAT.hdiv_trace import HDivTrace
-from FIAT.mixed import MixedElement                       # noqa: F401
-from FIAT.restricted import RestrictedElement             # noqa: F401
-from FIAT.quadrature_element import QuadratureElement     # noqa: F401
-from FIAT.kong_mulder_veldhuizen import KongMulderVeldhuizen  # noqa: F401
+from FIAT.kong_mulder_veldhuizen import KongMulderVeldhuizen
 from FIAT.fdm_element import FDMLagrange, FDMDiscontinuousLagrange, FDMQuadrature, FDMBrokenH1, FDMBrokenL2, FDMHermite  # noqa: F401
 
-# Important functionality
-from FIAT.quadrature import make_quadrature               # noqa: F401
-from FIAT.quadrature_schemes import create_quadrature     # noqa: F401
-from FIAT.reference_element import ufc_cell, ufc_simplex  # noqa: F401
-from FIAT.hdivcurl import Hdiv, Hcurl                     # noqa: F401
-
-__version__ = pkg_resources.get_distribution("fenics-fiat").version
-
 # List of supported elements and mapping to element classes
 supported_elements = {"Argyris": Argyris,
                       "Bell": Bell,
@@ -112,8 +110,11 @@
                       "BrokenElement": DiscontinuousElement,
                       "HDiv Trace": HDivTrace,
                       "Hellan-Herrmann-Johnson": HellanHerrmannJohnson,
+                      "Gopalakrishnan-Lederer-Schoberl 1st kind": GopalakrishnanLedererSchoberlFirstKind,
+                      "Gopalakrishnan-Lederer-Schoberl 2nd kind": GopalakrishnanLedererSchoberlSecondKind,
                       "Conforming Arnold-Winther": ArnoldWinther,
                       "Nonconforming Arnold-Winther": ArnoldWintherNC,
+                      "Hu-Zhang": HuZhang,
                       "Mardal-Tai-Winther": MardalTaiWinther}
 
 # List of extra elements

FIAT/arnold_winther.py

@@ -9,157 +9,125 @@
 # SPDX-License-Identifier:    LGPL-3.0-or-later
 
 
-from FIAT.finite_element import CiarletElement
-from FIAT.dual_set import DualSet
-from FIAT.polynomial_set import ONSymTensorPolynomialSet, ONPolynomialSet
-from FIAT.functional import (
-    PointwiseInnerProductEvaluation as InnerProduct,
-    FrobeniusIntegralMoment as FIM,
-    IntegralMomentOfTensorDivergence,
-    IntegralLegendreNormalNormalMoment,
-    IntegralLegendreNormalTangentialMoment)
-
-from FIAT.quadrature import make_quadrature
+from FIAT import finite_element, dual_set, polynomial_set
+from FIAT.reference_element import TRIANGLE
+from FIAT.quadrature_schemes import create_quadrature
+from FIAT.functional import (ComponentPointEvaluation,
+                             TensorBidirectionalIntegralMoment,
+                             IntegralMomentOfTensorDivergence,
+                             IntegralLegendreNormalNormalMoment,
+                             IntegralLegendreNormalTangentialMoment)
 
 import numpy
 
 
-class ArnoldWintherNCDual(DualSet):
-    def __init__(self, cell, degree=2):
-        if not degree == 2:
-            raise ValueError("Nonconforming Arnold-Winther elements are"
-                             "only defined for degree 2.")
-        dofs = []
-        dof_ids = {}
-        dof_ids[0] = {0: [], 1: [], 2: []}
-        dof_ids[1] = {0: [], 1: [], 2: []}
-        dof_ids[2] = {0: []}
+class ArnoldWintherNCDual(dual_set.DualSet):
+    def __init__(self, ref_el, degree=2):
+        if degree != 2:
+            raise ValueError("Nonconforming Arnold-Winther elements are only defined for degree 2.")
+        top = ref_el.get_topology()
+        sd = ref_el.get_spatial_dimension()
+        entity_ids = {dim: {entity: [] for entity in sorted(top[dim])} for dim in sorted(top)}
+        nodes = []
 
-        dof_cur = 0
         # no vertex dofs
         # proper edge dofs now (not the contraints)
-        # moments of normal . sigma against constants and linears.
-        for entity_id in range(3):                  # a triangle has 3 edges
-            for order in (0, 1):
-                dofs += [IntegralLegendreNormalNormalMoment(cell, entity_id, order, 6),
-                         IntegralLegendreNormalTangentialMoment(cell, entity_id, order, 6)]
-            dof_ids[1][entity_id] = list(range(dof_cur, dof_cur+4))
-            dof_cur += 4
+        # edge dofs: bidirectional nn and nt moments against P1.
+        qdegree = degree + 2
+        for entity in sorted(top[1]):
+            cur = len(nodes)
+            for order in range(2):
+                nodes.append(IntegralLegendreNormalNormalMoment(ref_el, entity, order, qdegree))
+                nodes.append(IntegralLegendreNormalTangentialMoment(ref_el, entity, order, qdegree))
+            entity_ids[1][entity].extend(range(cur, len(nodes)))
 
         # internal dofs: constant moments of three unique components
-        Q = make_quadrature(cell, 2)
-
-        e1 = numpy.array([1.0, 0.0])              # euclidean basis 1
-        e2 = numpy.array([0.0, 1.0])              # euclidean basis 2
-        basis = [(e1, e1), (e1, e2), (e2, e2)]    # basis for symmetric matrices
-        for (v1, v2) in basis:
-            v1v2t = numpy.outer(v1, v2)
-            fatqp = numpy.zeros((2, 2, len(Q.pts)))
-            for i, y in enumerate(v1v2t):
-                for j, x in enumerate(y):
-                    for k in range(len(Q.pts)):
-                        fatqp[i, j, k] = x
-            dofs.append(FIM(cell, Q, fatqp))
-        dof_ids[2][0] = list(range(dof_cur, dof_cur + 3))
-        dof_cur += 3
+        cur = len(nodes)
+        n = list(map(ref_el.compute_scaled_normal, sorted(top[sd-1])))
+        Q = create_quadrature(ref_el, degree)
+        phi = numpy.full(Q.get_weights().shape, 1/ref_el.volume())
+        nodes.extend(TensorBidirectionalIntegralMoment(ref_el, n[i+1], n[j+1], Q, phi)
+                     for i in range(sd) for j in range(i, sd))
+        entity_ids[2][0].extend(range(cur, len(nodes)))
 
         # put the constraint dofs last.
-        for entity_id in range(3):
-            dof = IntegralLegendreNormalNormalMoment(cell, entity_id, 2, 6)
-            dofs.append(dof)
-            dof_ids[1][entity_id].append(dof_cur)
-            dof_cur += 1
+        for entity in sorted(top[1]):
+            cur = len(nodes)
+            nodes.append(IntegralLegendreNormalNormalMoment(ref_el, entity, 2, qdegree))
+            entity_ids[1][entity].append(cur)
 
-        super().__init__(dofs, cell, dof_ids)
+        super().__init__(nodes, ref_el, entity_ids)
 
 
-class ArnoldWintherNC(CiarletElement):
+class ArnoldWintherNC(finite_element.CiarletElement):
     """The definition of the nonconforming Arnold-Winther element.
     """
-    def __init__(self, cell, degree=2):
-        assert degree == 2, "Only defined for degree 2"
-        Ps = ONSymTensorPolynomialSet(cell, degree)
-        Ls = ArnoldWintherNCDual(cell, degree)
+    def __init__(self, ref_el, degree=2):
+        if ref_el.shape != TRIANGLE:
+            raise ValueError(f"{type(self).__name__} only defined on triangles")
+        Ps = polynomial_set.ONSymTensorPolynomialSet(ref_el, degree)
+        Ls = ArnoldWintherNCDual(ref_el, degree)
+        formdegree = ref_el.get_spatial_dimension() - 1
         mapping = "double contravariant piola"
+        super().__init__(Ps, Ls, degree, formdegree, mapping=mapping)
 
-        super().__init__(Ps, Ls, degree, mapping=mapping)
 
-
-class ArnoldWintherDual(DualSet):
-    def __init__(self, cell, degree=3):
-        if not degree == 3:
-            raise ValueError("Arnold-Winther elements are"
-                             "only defined for degree 3.")
-        dofs = []
-        dof_ids = {}
-        dof_ids[0] = {0: [], 1: [], 2: []}
-        dof_ids[1] = {0: [], 1: [], 2: []}
-        dof_ids[2] = {0: []}
-
-        dof_cur = 0
+class ArnoldWintherDual(dual_set.DualSet):
+    def __init__(self, ref_el, degree=3):
+        if degree != 3:
+            raise ValueError("Arnold-Winther elements are only defined for degree 3.")
+        top = ref_el.get_topology()
+        sd = ref_el.get_spatial_dimension()
+        shp = (sd, sd)
+        entity_ids = {dim: {entity: [] for entity in sorted(top[dim])} for dim in sorted(top)}
+        nodes = []
 
         # vertex dofs
-        vs = cell.get_vertices()
-        e1 = numpy.array([1.0, 0.0])
-        e2 = numpy.array([0.0, 1.0])
-        basis = [(e1, e1), (e1, e2), (e2, e2)]
-
-        dof_cur = 0
-
-        for entity_id in range(3):
-            node = tuple(vs[entity_id])
-            for (v1, v2) in basis:
-                dofs.append(InnerProduct(cell, v1, v2, node))
-            dof_ids[0][entity_id] = list(range(dof_cur, dof_cur + 3))
-            dof_cur += 3
-
-        # edge dofs now
-        # moments of normal . sigma against constants and linears.
-        for entity_id in range(3):
-            for order in (0, 1):
-                dofs += [IntegralLegendreNormalNormalMoment(cell, entity_id, order, 6),
-                         IntegralLegendreNormalTangentialMoment(cell, entity_id, order, 6)]
-            dof_ids[1][entity_id] = list(range(dof_cur, dof_cur+4))
-            dof_cur += 4
-
-        # internal dofs: constant moments of three unique components
-        Q = make_quadrature(cell, 3)
-
-        e1 = numpy.array([1.0, 0.0])              # euclidean basis 1
-        e2 = numpy.array([0.0, 1.0])              # euclidean basis 2
-        basis = [(e1, e1), (e1, e2), (e2, e2)]    # basis for symmetric matrices
-        for (v1, v2) in basis:
-            v1v2t = numpy.outer(v1, v2)
-            fatqp = numpy.zeros((2, 2, len(Q.pts)))
-            for k in range(len(Q.pts)):
-                fatqp[:, :, k] = v1v2t
-            dofs.append(FIM(cell, Q, fatqp))
-        dof_ids[2][0] = list(range(dof_cur, dof_cur + 3))
-        dof_cur += 3
-
-        # Constraint dofs
-
-        Q = make_quadrature(cell, 5)
-
-        onp = ONPolynomialSet(cell, 2, (2,))
-        pts = Q.get_points()
-        onpvals = onp.tabulate(pts)[0, 0]
-
-        for i in list(range(3, 6)) + list(range(9, 12)):
-            dofs.append(IntegralMomentOfTensorDivergence(cell, Q,
-                                                         onpvals[i, :, :]))
-
-        dof_ids[2][0] += list(range(dof_cur, dof_cur+6))
-
-        super().__init__(dofs, cell, dof_ids)
-
-
-class ArnoldWinther(CiarletElement):
+        for v in sorted(top[0]):
+            cur = len(nodes)
+            pt, = ref_el.make_points(0, v, degree)
+            nodes.extend(ComponentPointEvaluation(ref_el, (i, j), shp, pt)
+                         for i in range(sd) for j in range(i, sd))
+            entity_ids[0][v].extend(range(cur, len(nodes)))
+
+        # edge dofs: bidirectional nn and nt moments against P_{k-2}
+        max_order = degree - 2
+        qdegree = degree + max_order
+        for entity in sorted(top[1]):
+            cur = len(nodes)
+            for order in range(max_order+1):
+                nodes.append(IntegralLegendreNormalNormalMoment(ref_el, entity, order, qdegree))
+                nodes.append(IntegralLegendreNormalTangentialMoment(ref_el, entity, order, qdegree))
+            entity_ids[1][entity].extend(range(cur, len(nodes)))
+
+        # internal dofs: moments of unique components against P_{k-3}
+        n = list(map(ref_el.compute_scaled_normal, sorted(top[sd-1])))
+        Q = create_quadrature(ref_el, 2*(degree-1))
+        P = polynomial_set.ONPolynomialSet(ref_el, degree-3, scale="L2 piola")
+        phis = P.tabulate(Q.get_points())[(0,)*sd]
+        nodes.extend(TensorBidirectionalIntegralMoment(ref_el, n[i+1], n[j+1], Q, phi)
+                     for phi in phis for i in range(sd) for j in range(i, sd))
+
+        # constraint dofs: moments of divergence against P_{k-1} \ P_{k-2}
+        P = polynomial_set.ONPolynomialSet(ref_el, degree-1, shape=(sd,))
+        dimPkm1 = P.expansion_set.get_num_members(degree-1)
+        dimPkm2 = P.expansion_set.get_num_members(degree-2)
+        PH = P.take([i + j * dimPkm1 for j in range(sd) for i in range(dimPkm2, dimPkm1)])
+        phis = PH.tabulate(Q.get_points())[(0,)*sd]
+        nodes.extend(IntegralMomentOfTensorDivergence(ref_el, Q, phi) for phi in phis)
+
+        entity_ids[2][0].extend(range(cur, len(nodes)))
+        super().__init__(nodes, ref_el, entity_ids)
+
+
+class ArnoldWinther(finite_element.CiarletElement):
     """The definition of the conforming Arnold-Winther element.
     """
-    def __init__(self, cell, degree=3):
-        assert degree == 3, "Only defined for degree 3"
-        Ps = ONSymTensorPolynomialSet(cell, degree)
-        Ls = ArnoldWintherDual(cell, degree)
+    def __init__(self, ref_el, degree=3):
+        if ref_el.shape != TRIANGLE:
+            raise ValueError(f"{type(self).__name__} only defined on triangles")
+        Ps = polynomial_set.ONSymTensorPolynomialSet(ref_el, degree)
+        Ls = ArnoldWintherDual(ref_el, degree)
+        formdegree = ref_el.get_spatial_dimension() - 1
         mapping = "double contravariant piola"
-        super().__init__(Ps, Ls, degree, mapping=mapping)
+        super().__init__(Ps, Ls, degree, formdegree, mapping=mapping)

FIAT/expansions.py

@@ -279,16 +279,19 @@ def __init__(self, ref_el, scale=None, variant=None):
         self._dmats_cache = {}
         self._cell_node_map_cache = {}
 
-    def get_scale(self, cell=0):
+    def get_scale(self, n, cell=0):
         scale = self.scale
+        sd = self.ref_el.get_spatial_dimension()
         if isinstance(scale, str):
-            sd = self.ref_el.get_spatial_dimension()
             vol = self.ref_el.volume_of_subcomplex(sd, cell)
             scale = scale.lower()
             if scale == "orthonormal":
                 scale = math.sqrt(1.0 / vol)
             elif scale == "l2 piola":
                 scale = 1.0 / vol
+        elif n == 0 and sd > 1 and len(self.affine_mappings) == 1:
+            # return 1 for n=0 to make regression tests pass
+            scale = 1
         return scale
 
     def get_num_members(self, n):
@@ -310,9 +313,7 @@ def _tabulate_on_cell(self, n, pts, order=0, cell=0, direction=None):
         ref_pts = numpy.add(numpy.dot(pts, A.T), b).T
         Jinv = A if direction is None else numpy.dot(A, direction)[:, None]
         sd = self.ref_el.get_spatial_dimension()
-
-        # Always return 1 for n=0 to make regression tests pass
-        scale = 1.0 if n == 0 and len(self.affine_mappings) == 1 else self.get_scale(cell=cell)
+        scale = self.get_scale(n, cell=cell)
         phi = dubiner_recurrence(sd, n, lorder, ref_pts, Jinv,
                                  scale, variant=self.variant)
         if self.continuity == "C0":
@@ -549,7 +550,7 @@ def _tabulate_on_cell(self, n, pts, order=0, cell=0, direction=None):
         Jinv = A[0, 0] if direction is None else numpy.dot(A, direction)
         xs = numpy.add(numpy.dot(pts, A.T), b)
         results = {}
-        scale = self.get_scale(cell=cell) * numpy.sqrt(2 * numpy.arange(n+1) + 1)
+        scale = self.get_scale(n, cell=cell) * numpy.sqrt(2 * numpy.arange(n+1) + 1)
         for k in range(order+1):
             v = numpy.zeros((n + 1, len(xs)), xs.dtype)
             if n >= k: