# FIAT package¶

## FIAT.P0 module¶

class FIAT.P0.P0(ref_el)
class FIAT.P0.P0Dual(ref_el)

## FIAT.argyris module¶

class FIAT.argyris.Argyris(ref_el, degree)

The Argyris finite element.

class FIAT.argyris.ArgyrisDualSet(ref_el, degree)
class FIAT.argyris.QuinticArgyris(ref_el)

The Argyris finite element.

class FIAT.argyris.QuinticArgyrisDualSet(ref_el)

## FIAT.bell module¶

class FIAT.bell.Bell(ref_el)

The Bell finite element.

class FIAT.bell.BellDualSet(ref_el)

## FIAT.brezzi_douglas_fortin_marini module¶

class FIAT.brezzi_douglas_fortin_marini.BDFMDualSet(ref_el, degree)
FIAT.brezzi_douglas_fortin_marini.BDFMSpace(ref_el, order)
class FIAT.brezzi_douglas_fortin_marini.BrezziDouglasFortinMarini(ref_el, degree)

The BDFM element

## FIAT.brezzi_douglas_marini module¶

class FIAT.brezzi_douglas_marini.BDMDualSet(ref_el, degree)
class FIAT.brezzi_douglas_marini.BrezziDouglasMarini(ref_el, degree)

The BDM element

## FIAT.bubble module¶

class FIAT.bubble.Bubble(ref_el, degree)

The bubble finite element: the dofs of the Lagrange FE in the interior of the cell

class FIAT.bubble.CodimBubble(ref_el, degree, codim)

Bubbles of a certain codimension.

class FIAT.bubble.FacetBubble(ref_el, degree)

The facet bubble finite element: the dofs of the Lagrange FE in the interior of the facets

## FIAT.crouzeix_raviart module¶

class FIAT.crouzeix_raviart.CrouzeixRaviart(cell, degree)

The Crouzeix-Raviart finite element:

K: Triangle/Tetrahedron Polynomial space: P_1 Dual basis: Evaluation at facet midpoints

class FIAT.crouzeix_raviart.CrouzeixRaviartDualSet(cell, degree)

Dual basis for Crouzeix-Raviart element (linears continuous at boundary midpoints).

## FIAT.discontinuous module¶

class FIAT.discontinuous.DiscontinuousElement(element)

A copy of an existing element where all dofs are associated with the cell

degree()

Return the degree of the (embedding) polynomial space.

dmats()

Return dmats: expansion coefficients for basis function derivatives.

get_coeffs()

Return the expansion coefficients for the basis of the finite element.

get_nodal_basis()

Return the nodal basis, encoded as a PolynomialSet object, for the finite element.

get_num_members(arg)

Return number of members of the expansion set.

get_order()

Return the order of the element (may be different from the degree)

get_reference_element()

Return the reference element for the finite element.

mapping()

Return a list of appropriate mappings from the reference element to a physical element for each basis function of the finite element.

num_sub_elements()

Return the number of sub-elements.

space_dimension()

Return the dimension of the finite element space.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to given order of basis functions at given points.

value_shape()

Return the value shape of the finite element functions.

## FIAT.discontinuous_lagrange module¶

FIAT.discontinuous_lagrange.DiscontinuousLagrange(ref_el, degree)
class FIAT.discontinuous_lagrange.DiscontinuousLagrangeDualSet(ref_el, degree)

The dual basis for Lagrange elements. This class works for simplices of any dimension. Nodes are point evaluation at equispaced points. This is the discontinuous version where all nodes are topologically associated with the cell itself

class FIAT.discontinuous_lagrange.HigherOrderDiscontinuousLagrange(ref_el, degree)

The discontinuous Lagrange finite element. It is what it is.

## FIAT.discontinuous_raviart_thomas module¶

class FIAT.discontinuous_raviart_thomas.DRTDualSet(ref_el, degree)

Dual basis for Raviart-Thomas elements consisting of point evaluation of normals on facets of codimension 1 and internal moments against polynomials. This is the discontinuous version where all nodes are topologically associated with the cell itself

class FIAT.discontinuous_raviart_thomas.DiscontinuousRaviartThomas(ref_el, q)

The discontinuous Raviart-Thomas finite element

## FIAT.discontinuous_taylor module¶

FIAT.discontinuous_taylor.DiscontinuousTaylor(ref_el, degree)
class FIAT.discontinuous_taylor.DiscontinuousTaylorDualSet(ref_el, degree)

The dual basis for Taylor elements. This class works for intervals. Nodes are function and derivative evaluation at the midpoint.

class FIAT.discontinuous_taylor.HigherOrderDiscontinuousTaylor(ref_el, degree)

The discontinuous Taylor finite element. Use a Taylor basis for DG.

## FIAT.dual_set module¶

class FIAT.dual_set.DualSet(nodes, ref_el, entity_ids)

Bases: object

get_entity_closure_ids()
get_entity_ids()
get_nodes()
get_reference_element()
to_riesz(poly_set)

## FIAT.enriched module¶

class FIAT.enriched.EnrichedElement(*elements)

Class implementing a finite element that combined the degrees of freedom of two existing finite elements.

This is an implementation which does not care about orthogonality of primal and dual basis.

degree()

Return the degree of the (embedding) polynomial space.

dmats()

Return dmats: expansion coefficients for basis function derivatives.

elements()

Return reference to original subelements

get_coeffs()

Return the expansion coefficients for the basis of the finite element.

get_nodal_basis()

Return the nodal basis, encoded as a PolynomialSet object, for the finite element.

get_num_members(arg)

Return number of members of the expansion set.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to given order of basis functions at given points.

value_shape()

Return the value shape of the finite element functions.

## FIAT.expansions module¶

Principal orthogonal expansion functions as defined by Karniadakis and Sherwin. These are parametrized over a reference element so as to allow users to get coordinates that they want.

class FIAT.expansions.LineExpansionSet(ref_el)

Bases: object

Evaluates the Legendre basis on a line reference element.

get_num_members(n)
tabulate(n, pts)

Returns a numpy array A[i,j] = phi_i(pts[j])

tabulate_derivatives(n, pts)

Returns a tuple of length one (A,) such that A[i,j] = D phi_i(pts[j]). The tuple is returned for compatibility with the interfaces of the triangle and tetrahedron expansions.

class FIAT.expansions.TetrahedronExpansionSet(ref_el)

Bases: object

Collapsed orthonormal polynomial expanion on a tetrahedron.

get_num_members(n)
tabulate(n, pts)
tabulate_derivatives(n, pts)
tabulate_jet(n, pts, order=1)
class FIAT.expansions.TriangleExpansionSet(ref_el)

Bases: object

Evaluates the orthonormal Dubiner basis on a triangular reference element.

get_num_members(n)
tabulate(n, pts)
tabulate_derivatives(n, pts)
tabulate_jet(n, pts, order=1)
FIAT.expansions.get_expansion_set(ref_el)

Returns an ExpansionSet instance appopriate for the given reference element.

FIAT.expansions.jrc(a, b, n)
FIAT.expansions.polynomial_dimension(ref_el, degree)

Returns the dimension of the space of polynomials of degree no greater than degree on the reference element.

FIAT.expansions.xi_tetrahedron(eta)

Maps from [-1,1]^3 to the -1/1 reference tetrahedron.

FIAT.expansions.xi_triangle(eta)

Maps from [-1,1]^2 to the (-1,1) reference triangle.

## FIAT.finite_element module¶

class FIAT.finite_element.CiarletElement(poly_set, dual, order, formdegree=None, mapping='affine')

Class implementing Ciarlet’s abstraction of a finite element being a domain, function space, and set of nodes.

Elements derived from this class are nodal finite elements, with a nodal basis generated from polynomials encoded in a PolynomialSet.

degree()

Return the degree of the (embedding) polynomial space.

dmats()

Return dmats: expansion coefficients for basis function derivatives.

get_coeffs()

Return the expansion coefficients for the basis of the finite element.

get_nodal_basis()

Return the nodal basis, encoded as a PolynomialSet object, for the finite element.

get_num_members(arg)

Return number of members of the expansion set.

static is_nodal()

True if primal and dual bases are orthogonal. If false, dual basis is not implemented or is undefined.

All implementations/subclasses are nodal including this one.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to given order of basis functions at given points.

Parameters: order – The maximum order of derivative. points – An iterable of points. entity – Optional (dimension, entity number) pair indicating which topological entity of the reference element to tabulate on. If None, default cell-wise tabulation is performed.
value_shape()

Return the value shape of the finite element functions.

class FIAT.finite_element.FiniteElement(ref_el, dual, order, formdegree=None, mapping='affine')

Bases: object

Class implementing a basic abstraction template for general finite element families. Finite elements which inherit from this class are non-nodal unless they are CiarletElement subclasses.

dual_basis()

Return the dual basis (list of functionals) for the finite element.

entity_closure_dofs()

Return the map of topological entities to degrees of freedom on the closure of those entities for the finite element.

entity_dofs()

Return the map of topological entities to degrees of freedom for the finite element.

get_dual_set()

Return the dual for the finite element.

get_formdegree()

Return the degree of the associated form (FEEC)

get_order()

Return the order of the element (may be different from the degree).

get_reference_element()

Return the reference element for the finite element.

static is_nodal()

True if primal and dual bases are orthogonal. If false, dual basis is not implemented or is undefined.

Subclasses may not necessarily be nodal, unless it is a CiarletElement.

mapping()

Return a list of appropriate mappings from the reference element to a physical element for each basis function of the finite element.

num_sub_elements()

Return the number of sub-elements.

space_dimension()

Return the dimension of the finite element space.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to given order of basis functions at given points.

Parameters: order – The maximum order of derivative. points – An iterable of points. entity – Optional (dimension, entity number) pair indicating which topological entity of the reference element to tabulate on. If None, default cell-wise tabulation is performed.
FIAT.finite_element.entity_support_dofs(elem, entity_dim)

Return the map of entity id to the degrees of freedom for which the corresponding basis functions take non-zero values

Parameters: elem – FIAT finite element entity_dim – Dimension of the cell subentity.

## FIAT.functional module¶

class FIAT.functional.ComponentPointEvaluation(ref_el, comp, shp, x)

Class representing point evaluation of a particular component of a vector function at a particular point x.

tostr()
class FIAT.functional.FrobeniusIntegralMoment(ref_el, Q, f_at_qpts)
class FIAT.functional.Functional(ref_el, target_shape, pt_dict, deriv_dict, functional_type)

Bases: object

Class implementing an abstract functional. All functionals are discrete in the sense that the are written as a weighted sum of (components of) their argument evaluated at particular points.

evaluate(f)

Obsolete and broken functional evaluation.

To evaluate the functional, call it on the target function:

functional(function)
get_point_dict()

Returns the functional information, which is a dictionary mapping each point in the support of the functional to a list of pairs containing the weight and component.

get_reference_element()

Returns the reference element.

get_type_tag()

Returns the type of function (e.g. point evaluation or normal component, which is probably handy for clients of FIAT

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

tostr()
class FIAT.functional.IntegralMoment(ref_el, Q, f_at_qpts, comp=(), shp=())

An IntegralMoment is a functional

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

class FIAT.functional.IntegralMomentOfNormalDerivative(ref_el, facet_no, Q, f_at_qpts)

Functional giving normal derivative integrated against some function on a facet.

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

class FIAT.functional.PointDerivative(ref_el, x, alpha)

Class representing point partial differentiation of scalar functions at a particular point x.

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

class FIAT.functional.PointEdgeTangentEvaluation(ref_el, edge_no, pt)

Implements the evaluation of the tangential component of a vector at a point on a facet of dimension 1.

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

tostr()
class FIAT.functional.PointEvaluation(ref_el, x)

Class representing point evaluation of scalar functions at a particular point x.

tostr()
class FIAT.functional.PointFaceTangentEvaluation(ref_el, face_no, tno, pt)

Implements the evaluation of a tangential component of a vector at a point on a facet of codimension 1.

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

tostr()
class FIAT.functional.PointNormalDerivative(ref_el, facet_no, pt)
to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

class FIAT.functional.PointNormalEvaluation(ref_el, facet_no, pt)

Implements the evaluation of the normal component of a vector at a point on a facet of codimension 1.

class FIAT.functional.PointScaledNormalEvaluation(ref_el, facet_no, pt)

Implements the evaluation of the normal component of a vector at a point on a facet of codimension 1, where the normal is scaled by the volume of that facet.

to_riesz(poly_set)

Constructs an array representation of the functional over the base of the given polynomial_set so that f(phi) for any phi in poly_set is given by a dot product.

tostr()
class FIAT.functional.PointwiseInnerProductEvaluation(ref_el, v, w, p)

This is a functional on symmetric 2-tensor fields. Let u be such a field, p be a point, and v,w be vectors. This implements the evaluation v^T u(p) w.

Clearly v^iu_{ij}w^j = u_{ij}v^iw^j. Thus the value can be computed from the Frobenius inner product of u with wv^T. This gives the correct weights.

FIAT.functional.index_iterator(shp)

Constructs a generator iterating over all indices in shp in generalized column-major order So if shp = (2,2), then we construct the sequence (0,0),(0,1),(1,0),(1,1)

## FIAT.gauss_legendre module¶

class FIAT.gauss_legendre.GaussLegendre(ref_el, degree)

1D discontinuous element with nodes at the Gauss-Legendre points.

class FIAT.gauss_legendre.GaussLegendreDualSet(ref_el, degree)

The dual basis for 1D discontinuous elements with nodes at the Gauss-Legendre points.

## FIAT.gauss_lobatto_legendre module¶

class FIAT.gauss_lobatto_legendre.GaussLobattoLegendre(ref_el, degree)

1D continuous element with nodes at the Gauss-Lobatto points.

class FIAT.gauss_lobatto_legendre.GaussLobattoLegendreDualSet(ref_el, degree)

The dual basis for 1D continuous elements with nodes at the Gauss-Lobatto points.

## FIAT.hdiv_trace module¶

class FIAT.hdiv_trace.HDivTrace(ref_el, degree)

Class implementing the trace of hdiv elements. This class is a stand-alone element family that produces a DG-facet field. This element is what’s produced after performing the trace operation on an existing H(Div) element.

This element is also known as the discontinuous trace field that arises in several DG formulations.

degree()

Return the degree of the (embedding) polynomial space.

dmats()

Return dmats: expansion coefficients for basis function derivatives.

get_coeffs()

Return the expansion coefficients for the basis of the finite element.

get_nodal_basis()

Return the nodal basis, encoded as a PolynomialSet object, for the finite element.

get_num_members(arg)

Return number of members of the expansion set.

static is_nodal()

True if primal and dual bases are orthogonal. If false, dual basis is not implemented or is undefined.

Subclasses may not necessarily be nodal, unless it is a CiarletElement.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to a given order of basis functions at given points.

Parameters: order – The maximum order of derivative. points – An iterable of points. entity – Optional (dimension, entity number) pair indicating which topological entity of the reference element to tabulate on. If None, tabulated values are computed by geometrically approximating which facet the points are on.

Note

Performing illegal tabulations on this element will result in either a tabulation table of numpy.nan arrays (entity=None case), or insertions of the TraceError exception class. This is due to the fact that performing cell-wise tabulations, or asking for any order of derivative evaluations, are not mathematically well-defined.

value_shape()

Return the value shape of the finite element functions.

exception FIAT.hdiv_trace.TraceError(msg)

Bases: Exception

Exception caused by tabulating a trace element on the interior of a cell, or the gradient of a trace element.

FIAT.hdiv_trace.barycentric_coordinates(points, vertices)

Computes the barycentric coordinates for a set of points relative to a simplex defined by a set of vertices.

Parameters: points – A set of points. vertices – A set of vertices that define the simplex.
FIAT.hdiv_trace.construct_dg_element(ref_el, degree)

Constructs a discontinuous galerkin element of a given degree on a particular reference cell.

FIAT.hdiv_trace.extract_unique_facet(coordinates, tolerance=1e-10)

Determines whether a set of points (described in its barycentric coordinates) are all on one of the facet sub-entities, and return the particular facet and whether the search has been successful.

Parameters: coordinates – A set of points described in barycentric coordinates. tolerance – A fixed tolerance for geometric identifications.
FIAT.hdiv_trace.map_from_reference_facet(point, vertices)

Evaluates the physical coordinate of a point using barycentric coordinates.

Parameters: point – The reference points to be mapped to the facet. vertices – The vertices defining the physical element.
FIAT.hdiv_trace.map_to_reference_facet(points, vertices, facet)

Given a set of points and vertices describing a facet of a simplex in n-dimensional coordinates (where the points lie on the facet), map the points to the reference simplex of dimension (n-1).

Parameters: points – A set of points in n-D. vertices – A set of vertices describing a facet of a simplex in n-D. facet – Integer representing the facet number.

## FIAT.hdivcurl module¶

FIAT.hdivcurl.Hcurl(element)
FIAT.hdivcurl.Hdiv(element)

## FIAT.hellan_herrmann_johnson module¶

Implementation of the Hellan-Herrmann-Johnson finite elements.

class FIAT.hellan_herrmann_johnson.HellanHerrmannJohnson(cell, degree)

The definition of Hellan-Herrmann-Johnson element. It is defined only in dimension 2. It consists of piecewise polynomial symmetric-matrix-valued functions of degree r or less with normal-normal continuity.

class FIAT.hellan_herrmann_johnson.HellanHerrmannJohnsonDual(cell, degree)

Degrees of freedom for Hellan-Herrmann-Johnson elements.

## FIAT.hermite module¶

class FIAT.hermite.CubicHermite(ref_el)

The cubic Hermite finite element. It is what it is.

class FIAT.hermite.CubicHermiteDualSet(ref_el)

The dual basis for Lagrange elements. This class works for simplices of any dimension. Nodes are point evaluation at equispaced points.

## FIAT.jacobi module¶

Several functions related to the one-dimensional jacobi polynomials: Evaluation, evaluation of derivatives, plus computation of the roots via Newton’s method. These mainly are used in defining the expansion functions over the simplices and in defining quadrature rules over each domain.

FIAT.jacobi.eval_jacobi(a, b, n, x)

Evaluates the nth jacobi polynomial with weight parameters a,b at a point x. Recurrence relations implemented from the pseudocode given in Karniadakis and Sherwin, Appendix B

FIAT.jacobi.eval_jacobi_batch(a, b, n, xs)

Evaluates all jacobi polynomials with weights a,b up to degree n. xs is a numpy.array of points. Returns a two-dimensional array of points, where the rows correspond to the Jacobi polynomials and the columns correspond to the points.

FIAT.jacobi.eval_jacobi_deriv(a, b, n, x)

Evaluates the first derivative of P_{n}^{a,b} at a point x.

FIAT.jacobi.eval_jacobi_deriv_batch(a, b, n, xs)

Evaluates the first derivatives of all jacobi polynomials with weights a,b up to degree n. xs is a numpy.array of points. Returns a two-dimensional array of points, where the rows correspond to the Jacobi polynomials and the columns correspond to the points.

## FIAT.lagrange module¶

class FIAT.lagrange.Lagrange(ref_el, degree)

The Lagrange finite element. It is what it is.

class FIAT.lagrange.LagrangeDualSet(ref_el, degree)

The dual basis for Lagrange elements. This class works for simplices of any dimension. Nodes are point evaluation at equispaced points.

## FIAT.mixed module¶

class FIAT.mixed.MixedElement(elements, ref_el=None)

A FIAT-like representation of a mixed element.

Parameters: elements – An iterable of FIAT elements. ref_el – The reference element (optional).

This object offers tabulation of the concatenated basis function tables along with an entity_dofs dict.

elements()
get_nodal_basis()
is_nodal()

True if primal and dual bases are orthogonal.

mapping()

Return a list of appropriate mappings from the reference element to a physical element for each basis function of the finite element.

num_sub_elements()

Return the number of sub-elements.

tabulate(order, points, entity=None)

Tabulate a mixed element by appropriately splatting together the tabulation of the individual elements.

value_shape()
FIAT.mixed.concatenate_entity_dofs(ref_el, elements)

Combine the entity_dofs from a list of elements into a combined entity_dof containing the information for the concatenated DoFs of all the elements.

## FIAT.morley module¶

class FIAT.morley.Morley(ref_el)

The Morley finite element.

class FIAT.morley.MorleyDualSet(ref_el)

The dual basis for Lagrange elements. This class works for simplices of any dimension. Nodes are point evaluation at equispaced points.

## FIAT.nedelec module¶

class FIAT.nedelec.Nedelec(ref_el, q)

Nedelec finite element

class FIAT.nedelec.NedelecDual2D(ref_el, degree)

Dual basis for first-kind Nedelec in 2d

class FIAT.nedelec.NedelecDual3D(ref_el, degree)

Dual basis for first-kind Nedelec in 3d

FIAT.nedelec.NedelecSpace2D(ref_el, k)

Constructs a basis for the 2d H(curl) space of the first kind which is (P_k)^2 + P_k rot( x )

FIAT.nedelec.NedelecSpace3D(ref_el, k)

Constructs a nodal basis for the 3d first-kind Nedelec space

## FIAT.nedelec_second_kind module¶

class FIAT.nedelec_second_kind.NedelecSecondKind(cell, degree)

The H(curl) Nedelec elements of the second kind on triangles and tetrahedra: the polynomial space described by the full polynomials of degree k, with a suitable set of degrees of freedom to ensure H(curl) conformity.

class FIAT.nedelec_second_kind.NedelecSecondKindDual(cell, degree)

This class represents the dual basis for the Nedelec H(curl) elements of the second kind. The degrees of freedom (L) for the elements of the k’th degree are

d = 2:

vertices: None

edges: L(f) = f (x_i) * t for (k+1) points x_i on each edge

cell: L(f) = int f * g * dx for g in RT_{k-1}

d = 3:

vertices: None

edges: L(f) = f(x_i) * t for (k+1) points x_i on each edge

faces: L(f) = int_F f * g * ds for g in RT_{k-1}(F) for each face F

cell: L(f) = int f * g * dx for g in RT_{k-2}

Higher spatial dimensions are not yet implemented. (For d = 1, these elements coincide with the CG_k elements.)

generate_degrees_of_freedom(cell, degree)

Generate dofs and geometry-to-dof maps (ids).

## FIAT.nodal_enriched module¶

class FIAT.nodal_enriched.NodalEnrichedElement(*elements)

NodalEnriched element is a direct sum of a sequence of finite elements. Dual basis is reorthogonalized to the primal basis for nodality.

The following is equivalent:
• the constructor is well-defined,
• the resulting element is unisolvent and its basis is nodal,
• the supplied elements are unisolvent with nodal basis and their primal bases are mutually linearly independent,
• the supplied elements are unisolvent with nodal basis and their dual bases are mutually linearly independent.

## FIAT.orthopoly module¶

orthopoly.py - A suite of functions for generating orthogonal polynomials and quadrature rules.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Last updated on Wed Jan 1 14:29:25 MST 2014

Modified by David A. Ham (david.ham@imperial.ac.uk), 2016

FIAT.orthopoly.gauss(alpha, beta)

Compute the Gauss nodes and weights from the recursion coefficients associated with a set of orthogonal polynomials

Inputs: alpha - recursion coefficients beta - recursion coefficients

Adapted from the MATLAB code by Walter Gautschi http://www.cs.purdue.edu/archives/2002/wxg/codes/gauss.m

FIAT.orthopoly.jacobi(N, a, b, x, NOPT=1)

JACOBI computes the Jacobi polynomials which are orthogonal on [-1,1] with respect to the weight w(x)=[(1-x)^a]*[(1+x)^b] and evaluate them on the given grid up to P_N(x). Setting NOPT=2 returns the L2-normalized polynomials

FIAT.orthopoly.jacobiD(N, a, b, x, NOPT=1)

JACOBID computes the first derivatives of the normalized Jacobi polynomials which are orthogonal on [-1,1] with respect to the weight w(x)=[(1-x)^a]*[(1+x)^b] and evaluate them on the given grid up to P_N(x). Setting NOPT=2 returns the derivatives of the L2-normalized polynomials

FIAT.orthopoly.lobatto(alpha, beta, xl1, xl2)

Compute the Lobatto nodes and weights with the preassigned nodea xl1,xl2

Inputs: alpha - recursion coefficients beta - recursion coefficients xl1 - assigned node location xl2 - assigned node location

Based on the section 7 of the paper “Some modified matrix eigenvalue problems” by Gene Golub, SIAM Review Vol 15, No. 2, April 1973, pp.318–334

FIAT.orthopoly.mm_log(N, a)

MM_LOG Modified moments for a logarithmic weight function.

The call mm=MM_LOG(n,a) computes the first n modified moments of the logarithmic weight function w(t)=t^a log(1/t) on [0,1] relative to shifted Legendre polynomials.

REFERENCE: Walter Gautschi,On the preceding paper A Legendre
polynomial integral’ by James L. Blue’‘, Math. Comp. 33 (1979), 742-743.

Adapted from the MATLAB implementation: https://www.cs.purdue.edu/archives/2002/wxg/codes/mm_log.m

FIAT.orthopoly.mod_chebyshev(N, mom, alpham, betam)

Calcuate the recursion coefficients for the orthogonal polynomials which are are orthogonal with respect to a weight function which is represented in terms of its modifed moments which are obtained by integrating the monic polynomials against the weight function.

REFERENCES:

John C. Wheeler, “Modified moments and Gaussian quadratures” Rocky Mountain Journal of Mathematics, Vol. 4, Num. 2 (1974), 287–296

Walter Gautschi, “Orthogonal Polynomials (in Matlab) Journal of Computational and Applied Mathematics, Vol. 178 (2005) 215–234

Adapted from the MATLAB implementation: https://www.cs.purdue.edu/archives/2002/wxg/codes/chebyshev.m

FIAT.orthopoly.polyval(alpha, beta, x)

Evaluate polynomials on x given the recursion coefficients alpha and beta

FIAT.orthopoly.rec_jaclog(N, a)

Generate the recursion coefficients alpha_k, beta_k

P_{k+1}(x) = (x-alpha_k)*P_{k}(x) - beta_k P_{k-1}(x)

for the monic polynomials which are orthogonal on [0,1] with respect to the weight w(x)=x^a*log(1/x)

Inputs: N - polynomial order a - weight parameter

Outputs: alpha - recursion coefficients beta - recursion coefficients

Adated from the MATLAB code: https://www.cs.purdue.edu/archives/2002/wxg/codes/r_jaclog.m

FIAT.orthopoly.rec_jacobi(N, a, b)

Generate the recursion coefficients alpha_k, beta_k

P_{k+1}(x) = (x-alpha_k)*P_{k}(x) - beta_k P_{k-1}(x)

for the Jacobi polynomials which are orthogonal on [-1,1] with respect to the weight w(x)=[(1-x)^a]*[(1+x)^b]

Inputs: N - polynomial order a - weight parameter b - weight parameter

Outputs: alpha - recursion coefficients beta - recursion coefficients

Adapted from the MATLAB code by Dirk Laurie and Walter Gautschi http://www.cs.purdue.edu/archives/2002/wxg/codes/r_jacobi.m

FIAT.orthopoly.rec_jacobi01(N, a, b)

Generate the recursion coefficients alpha_k, beta_k for the Jacobi polynomials which are orthogonal on [0,1]

See rec_jacobi for the recursion coefficients on [-1,1]

Inputs: N - polynomial order a - weight parameter b - weight parameter

Outputs: alpha - recursion coefficients beta - recursion coefficients

Adapted from the MATLAB implementation: https://www.cs.purdue.edu/archives/2002/wxg/codes/r_jacobi01.m

## FIAT.polynomial_set module¶

class FIAT.polynomial_set.ONPolynomialSet(ref_el, degree, shape=())

Constructs an orthonormal basis out of expansion set by having an identity matrix of coefficients. Can be used to specify ON bases for vector- and tensor-valued sets as well.

class FIAT.polynomial_set.ONSymTensorPolynomialSet(ref_el, degree, size=None)

Constructs an orthonormal basis for symmetric-tensor-valued polynomials on a reference element.

class FIAT.polynomial_set.PolynomialSet(ref_el, degree, embedded_degree, expansion_set, coeffs, dmats)

Bases: object

Implements a set of polynomials as linear combinations of an expansion set over a reference element. ref_el: the reference element degree: an order labeling the space embedded degree: the degree of polynomial expansion basis that

must be used to evaluate this space
coeffs: A numpy array containing the coefficients of the expansion
basis for each member of the set. Coeffs is ordered by coeffs[i,j,k] where i is the label of the member, k is the label of the expansion function, and j is a (possibly empty) tuple giving the index for a vector- or tensor-valued function.
get_coeffs()
get_degree()
get_dmats()
get_embedded_degree()
get_expansion_set()
get_num_members()
get_reference_element()
get_shape()

Returns the shape of phi(x), where () corresponds to scalar (2,) a vector of length 2, etc

tabulate(pts, jet_order=0)

Returns the values of the polynomial set.

tabulate_new(pts)
take(items)

Extracts subset of polynomials given by items.

FIAT.polynomial_set.form_matrix_product(mats, alpha)

forms product over mats[i]**alpha[i]

FIAT.polynomial_set.mis(m, n)

returns all m-tuples of nonnegative integers that sum up to n.

FIAT.polynomial_set.polynomial_set_union_normalized(A, B)

Given polynomial sets A and B, constructs a new polynomial set whose span is the same as that of span(A) union span(B). It may not contain any of the same members of the set, as we construct a span via SVD.

FIAT.polynomial_set.project(f, U, Q)

Computes the expansion coefficients of f in terms of the members of a polynomial set U. Numerical integration is performed by quadrature rule Q.

class FIAT.quadrature.CollapsedQuadratureTetrahedronRule(ref_el, m)

Implements the collapsed quadrature rules defined in Karniadakis & Sherwin by mapping products of Gauss-Jacobi rules from the cube to the tetrahedron.

class FIAT.quadrature.CollapsedQuadratureTriangleRule(ref_el, m)

Implements the collapsed quadrature rules defined in Karniadakis & Sherwin by mapping products of Gauss-Jacobi rules from the square to the triangle.

class FIAT.quadrature.GaussJacobiQuadratureLineRule(ref_el, m)

Gauss-Jacobi quadature rule determined by Jacobi weights a and b using m roots of m:th order Jacobi polynomial.

class FIAT.quadrature.GaussLegendreQuadratureLineRule(ref_el, m)

Produce the Gauss–Legendre quadrature rules on the interval using the implementation in numpy. This facilitates implementing discontinuous spectral elements.

The quadrature rule uses m points for a degree of precision of 2m-1.

class FIAT.quadrature.GaussLobattoLegendreQuadratureLineRule(ref_el, m)

Implement the Gauss-Lobatto-Legendre quadrature rules on the interval using Greg von Winckel’s implementation. This facilitates implementing spectral elements.

The quadrature rule uses m points for a degree of precision of 2m-3.

class FIAT.quadrature.QuadratureRule(ref_el, pts, wts)

Bases: object

General class that models integration over a reference element as the weighted sum of a function evaluated at a set of points.

get_points()
get_weights()
integrate(f)
class FIAT.quadrature.UFCTetrahedronFaceQuadratureRule(face_number, degree)

Highly specialized quadrature rule for the face of a tetrahedron, mapped from a reference triangle, used for higher order Nedelecs

jacobian()
reference_rule()
FIAT.quadrature.compute_gauss_jacobi_points(a, b, m)

Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton’s method. The initial guesses are the Chebyshev points. Algorithm implemented in Python from the pseudocode given by Karniadakis and Sherwin

FIAT.quadrature.compute_gauss_jacobi_rule(a, b, m)
FIAT.quadrature.make_quadrature(ref_el, m)

Returns the collapsed quadrature rule using m points per direction on the given reference element. In the tensor product case, m is a tuple.

FIAT.quadrature.make_tensor_product_quadrature(*quad_rules)

Returns the quadrature rule for a TensorProduct cell, by combining the quadrature rules of the components.

class FIAT.quadrature_element.QuadratureElement(ref_el, points)

A set of quadrature points pretending to be a finite element.

static is_nodal()

True if primal and dual bases are orthogonal. If false, dual basis is not implemented or is undefined.

Subclasses may not necessarily be nodal, unless it is a CiarletElement.

tabulate(order, points, entity=None)

Return the identity matrix of size (num_quad_points, num_quad_points), in a format that monomialintegration and monomialtabulation understands.

value_shape()

This module generates quadrature schemes on reference cells that integrate exactly a polynomial of a given degree using a specified scheme.

Scheme options are:

scheme=”default”

scheme=”canonical” (collapsed Gauss scheme)

Background on the schemes:

Keast rules for tetrahedra:
Keast, P. Moderate-degree tetrahedral quadrature formulas, Computer Methods in Applied Mechanics and Engineering 55(3):339-348, 1986. http://dx.doi.org/10.1016/0045-7825(86)90059-9
FIAT.quadrature_schemes.create_quadrature(ref_el, degree, scheme='default')

Generate quadrature rule for given reference element that will integrate an polynomial of order ‘degree’ exactly.

For low-degree (<=6) polynomials on triangles and tetrahedra, this uses hard-coded rules, otherwise it falls back to a collapsed Gauss scheme on simplices. On tensor-product cells, it is a tensor-product quadrature rule of the subcells.

Parameters: cell – The FIAT cell to create the quadrature for. degree – The degree of polynomial that the rule should integrate exactly.

## FIAT.raviart_thomas module¶

class FIAT.raviart_thomas.RTDualSet(ref_el, degree)

Dual basis for Raviart-Thomas elements consisting of point evaluation of normals on facets of codimension 1 and internal moments against polynomials

FIAT.raviart_thomas.RTSpace(ref_el, deg)

Constructs a basis for the the Raviart-Thomas space (P_k)^d + P_k x

class FIAT.raviart_thomas.RaviartThomas(ref_el, q)

The Raviart-Thomas finite element

## FIAT.reference_element module¶

Abstract class and particular implementations of finite element reference simplex geometry/topology.

Provides an abstract base class and particular implementations for the reference simplex geometry and topology. The rest of FIAT is abstracted over this module so that different reference element geometry (e.g. a vertex at (0,0) versus at (-1,-1)) and orderings of entities have a single point of entry.

Currently implemented are UFC and Default Line, Triangle and Tetrahedron.

class FIAT.reference_element.Cell(shape, vertices, topology)

Bases: object

Abstract class for a reference cell. Provides accessors for geometry (vertex coordinates) as well as topology (orderings of vertices that make up edges, facecs, etc.

construct_subelement(dimension)

Constructs the reference element of a cell subentity specified by subelement dimension.

Parameters: dimension – tuple for tensor product cells, int otherwise
get_connectivity()

Returns a dictionary encoding the connectivity of the element.

The dictionary’s keys are the spatial dimensions pairs ((1, 0), (2, 0), (2, 1), …) and each value is a list with entities of second dimension ordered by local dim0-dim1 numbering.

get_dimension()

Returns the subelement dimension of the cell. For tensor product cells, this a tuple of dimensions for each cell in the product. For all other cells, this is the same as the spatial dimension.

get_entity_transform(dim, entity_i)

Returns a mapping of point coordinates from the entity_i-th subentity of dimension dim to the cell.

Parameters: dim – tuple for tensor product cells, int otherwise entity_i – entity number (integer)
get_shape()

Returns the code for the element’s shape.

get_spatial_dimension()

Returns the spatial dimension in which the element lives.

get_topology()

Returns a dictionary encoding the topology of the element.

The dictionary’s keys are the spatial dimensions (0, 1, …) and each value is a dictionary mapping.

get_vertices()

Returns an iterable of the element’s vertices, each stored as a tuple.

get_vertices_of_subcomplex(t)

Returns the tuple of vertex coordinates associated with the labels contained in the iterable t.

class FIAT.reference_element.DefaultLine

This is the reference line with vertices (-1.0,) and (1.0,).

get_facet_element()
class FIAT.reference_element.DefaultTetrahedron

This is the reference tetrahedron with vertices (-1,-1,-1), (1,-1,-1),(-1,1,-1), and (-1,-1,1).

get_facet_element()
class FIAT.reference_element.DefaultTriangle

This is the reference triangle with vertices (-1.0,-1.0), (1.0,-1.0), and (-1.0,1.0).

get_facet_element()
class FIAT.reference_element.IntrepidTetrahedron

This is the reference tetrahedron with vertices (0,0,0), (1,0,0),(0,1,0), and (0,0,1) used in the Intrepid project.

get_facet_element()
class FIAT.reference_element.IntrepidTriangle

This is the Intrepid triangle with vertices (0,0),(1,0),(0,1)

get_facet_element()
class FIAT.reference_element.Point

This is the reference point.

FIAT.reference_element.ReferenceElement
class FIAT.reference_element.Simplex(shape, vertices, topology)

Abstract class for a reference simplex.

compute_edge_tangent(edge_i)

Computes the nonnormalized tangent to a 1-dimensional facet. returns a single vector.

compute_face_edge_tangents(dim, entity_id)

Computes all the edge tangents of any k-face with k>=1. The result is a array of binom(dim+1,2) vectors. This agrees with compute_edge_tangent when dim=1.

compute_face_tangents(face_i)

Computes the two tangents to a face. Only implemented for a tetrahedron.

compute_normal(facet_i)

Returns the unit normal vector to facet i of codimension 1.

compute_normalized_edge_tangent(edge_i)

Computes the unit tangent vector to a 1-dimensional facet

compute_normalized_tangents(dim, i)

computes tangents in any dimension based on differences between vertices and the first vertex of the i:th facet of dimension dim. Returns a (possibly empty) list. These tangents are normalized to have unit length.

compute_reference_normal(facet_dim, facet_i)

Returns the unit normal in infinity norm to facet_i.

compute_scaled_normal(facet_i)

Returns the unit normal to facet_i of scaled by the volume of that facet.

compute_tangents(dim, i)

computes tangents in any dimension based on differences between vertices and the first vertex of the i:th facet of dimension dim. Returns a (possibly empty) list. These tangents are NOT normalized to have unit length.

get_dimension()

Returns the subelement dimension of the cell. Same as the spatial dimension.

get_entity_transform(dim, entity)

Returns a mapping of point coordinates from the entity-th subentity of dimension dim to the cell.

Parameters: dim – subentity dimension (integer) entity – entity number (integer)
make_lattice(n, interior=0)

Constructs a lattice of points on the simplex. For example, the 1:st order lattice will be just the vertices. The optional argument interior specifies how many points from the boundary to omit. For example, on a line with n = 2, and interior = 0, this function will return the vertices and midpoint, but with interior = 1, it will only return the midpoint.

make_points(dim, entity_id, order)

Constructs a lattice of points on the entity_id:th facet of dimension dim. Order indicates how many points to include in each direction.

volume()

Computes the volume of the simplex in the appropriate dimensional measure.

volume_of_subcomplex(dim, facet_no)
class FIAT.reference_element.TensorProductCell(*cells)

A cell that is the product of FIAT cells.

compute_reference_normal(facet_dim, facet_i)

Returns the unit normal in infinity norm to facet_i of subelement dimension facet_dim.

construct_subelement(dimension)

Constructs the reference element of a cell subentity specified by subelement dimension.

Parameters: dimension – dimension in each “direction” (tuple)
contains_point(point, epsilon=0)

Checks if reference cell contains given point (with numerical tolerance).

get_dimension()

Returns the subelement dimension of the cell, a tuple of dimensions for each cell in the product.

get_entity_transform(dim, entity_i)

Returns a mapping of point coordinates from the entity_i-th subentity of dimension dim to the cell.

Parameters: dim – subelement dimension (tuple) entity_i – entity number (integer)
volume()

Computes the volume in the appropriate dimensional measure.

class FIAT.reference_element.UFCHexahedron

This is the reference hexahedron with vertices (0.0, 0.0, 0.0), (0.0, 0.0, 1.0), (0.0, 1.0, 0.0), (0.0, 1.0, 1.0), (1.0, 0.0, 0.0), (1.0, 0.0, 1.0), (1.0, 1.0, 0.0) and (1.0, 1.0, 1.0).

compute_reference_normal(facet_dim, facet_i)

Returns the unit normal in infinity norm to facet_i.

construct_subelement(dimension)

Constructs the reference element of a cell subentity specified by subelement dimension.

Parameters: dimension – subentity dimension (integer)
contains_point(point, epsilon=0)

Checks if reference cell contains given point (with numerical tolerance).

get_dimension()

Returns the subelement dimension of the cell. Same as the spatial dimension.

get_entity_transform(dim, entity_i)

Returns a mapping of point coordinates from the entity_i-th subentity of dimension dim to the cell.

Parameters: dim – entity dimension (integer) entity_i – entity number (integer)
volume()

Computes the volume in the appropriate dimensional measure.

class FIAT.reference_element.UFCInterval

This is the reference interval with vertices (0.0,) and (1.0,).

class FIAT.reference_element.UFCQuadrilateral

This is the reference quadrilateral with vertices (0.0, 0.0), (0.0, 1.0), (1.0, 0.0) and (1.0, 1.0).

compute_reference_normal(facet_dim, facet_i)

Returns the unit normal in infinity norm to facet_i.

construct_subelement(dimension)

Constructs the reference element of a cell subentity specified by subelement dimension.

Parameters: dimension – subentity dimension (integer)
contains_point(point, epsilon=0)

Checks if reference cell contains given point (with numerical tolerance).

get_dimension()

Returns the subelement dimension of the cell. Same as the spatial dimension.

get_entity_transform(dim, entity_i)

Returns a mapping of point coordinates from the entity_i-th subentity of dimension dim to the cell.

Parameters: dim – entity dimension (integer) entity_i – entity number (integer)
volume()

Computes the volume in the appropriate dimensional measure.

class FIAT.reference_element.UFCSimplex(shape, vertices, topology)
construct_subelement(dimension)

Constructs the reference element of a cell subentity specified by subelement dimension.

Parameters: dimension – subentity dimension (integer)
contains_point(point, epsilon=0)

Checks if reference cell contains given point (with numerical tolerance).

get_facet_element()
class FIAT.reference_element.UFCTetrahedron

This is the reference tetrahedron with vertices (0,0,0), (1,0,0),(0,1,0), and (0,0,1).

compute_normal(i)

UFC consistent normals.

class FIAT.reference_element.UFCTriangle

This is the reference triangle with vertices (0.0,0.0), (1.0,0.0), and (0.0,1.0).

compute_normal(i)

UFC consistent normal

FIAT.reference_element.compute_unflattening_map(topology_dict)

This function returns unflattening map for the given tensor product topology dict.

FIAT.reference_element.default_simplex(spatial_dim)

Factory function that maps spatial dimension to an instance of the default reference simplex of that dimension.

FIAT.reference_element.flatten_entities(topology_dict)

This function flattens topology dict of TensorProductCell and entity_dofs dict of TensorProductElement

FIAT.reference_element.lattice_iter(start, finish, depth)

Generator iterating over the depth-dimensional lattice of integers between start and (finish-1). This works on simplices in 1d, 2d, 3d, and beyond

FIAT.reference_element.linalg_subspace_intersection(A, B)

Computes the intersection of the subspaces spanned by the columns of 2-dimensional arrays A,B using the algorithm found in Golub and van Loan (3rd ed) p. 604. A should be in R^{m,p} and B should be in R^{m,q}. Returns an orthonormal basis for the intersection of the spaces, stored in the columns of the result.

FIAT.reference_element.make_affine_mapping(xs, ys)

Constructs (A,b) such that x –> A * x + b is the affine mapping from the simplex defined by xs to the simplex defined by ys.

FIAT.reference_element.tuple_sum(tree)

This function calculates the sum of elements in a tuple, it is needed to handle nested tuples in TensorProductCell. Example: tuple_sum(((1, 0), 1)) returns 2 If input argument is not the tuple, returns input.

FIAT.reference_element.ufc_cell(cell)

Handle incoming calls from FFC.

FIAT.reference_element.ufc_simplex(spatial_dim)

Factory function that maps spatial dimension to an instance of the UFC reference simplex of that dimension.

FIAT.reference_element.volume(verts)

Constructs the volume of the simplex spanned by verts

## FIAT.regge module¶

Implementation of the generalized Regge finite elements.

class FIAT.regge.Regge(cell, degree)

The generalized Regge elements for symmetric-matrix-valued functions. REG(r) in dimension n is the space of polynomial symmetric-matrix-valued functions of degree r or less with tangential-tangential continuity.

class FIAT.regge.ReggeDual(cell, degree)

Degrees of freedom for generalized Regge finite elements.

## FIAT.restricted module¶

class FIAT.restricted.RestrictedElement(element, indices=None, restriction_domain=None)

Restrict given element to specified list of dofs.

FIAT.restricted.sorted_by_key(mapping)

Sort dict items by key, allowing different key types.

## FIAT.tensor_product module¶

class FIAT.tensor_product.FlattenedDimensions(element)

A wrapper class that flattens entity dimensions of a FIAT element defined on a TensorProductCell to one with quadrilateral/hexahedron entities. TensorProductCell has dimension defined as a tuple of factor element dimensions (i, j) in 2D and (i, j, k) in 3D. Flattened dimension is a sum of the tuple elements.

degree()

Return the degree of the (embedding) polynomial space.

dmats()

Return dmats: expansion coefficients for basis function derivatives.

get_coeffs()

Return the expansion coefficients for the basis of the finite element.

get_nodal_basis()

Return the nodal basis, encoded as a PolynomialSet object, for the finite element.

get_num_members(arg)

Return number of members of the expansion set.

is_nodal()

True if primal and dual bases are orthogonal. If false, dual basis is not implemented or is undefined.

Subclasses may not necessarily be nodal, unless it is a CiarletElement.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to given order of basis functions at given points.

value_shape()

Return the value shape of the finite element functions.

class FIAT.tensor_product.TensorProductElement(A, B)

Class implementing a finite element that is the tensor product of two existing finite elements.

degree()

Return the degree of the (embedding) polynomial space.

dmats()

Return dmats: expansion coefficients for basis function derivatives.

get_coeffs()

Return the expansion coefficients for the basis of the finite element.

get_nodal_basis()

Return the nodal basis, encoded as a PolynomialSet object, for the finite element.

get_num_members(arg)

Return number of members of the expansion set.

is_nodal()

True if primal and dual bases are orthogonal. If false, dual basis is not implemented or is undefined.

Subclasses may not necessarily be nodal, unless it is a CiarletElement.

tabulate(order, points, entity=None)

Return tabulated values of derivatives up to given order of basis functions at given points.

value_shape`()

Return the value shape of the finite element functions.

## Module contents¶

FInite element Automatic Tabulator – supports constructing and evaluating arbitrary order Lagrange and many other elements. Simplices in one, two, and three dimensions are supported.