Interface class for parametric scalar valued finite elements. More...
#include <lf/fe/fe.h>
Public Types | |
| using | Scalar = SCALAR |
| The underlying scalar type. | |
Public Member Functions | |
| virtual | ~ScalarReferenceFiniteElement ()=default |
| virtual base::RefEl | RefEl () const =0 |
| Tells the type of reference cell underlying the parametric finite element. | |
| virtual unsigned int | Degree () const =0 |
| Request the maximal polynomial degree of the basis functions in this finite element. | |
| dim_t | Dimension () const |
| Returns the spatial dimension of the reference cell. | |
| virtual size_type | NumRefShapeFunctions () const |
| Total number of reference shape functions associated with the reference cell. | |
| virtual size_type | NumRefShapeFunctions (dim_t codim) const |
| The number of interior reference shape functions for sub-entities of a particular co-dimension. | |
| virtual size_type | NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const =0 |
| The number of interior reference shape functions for every sub-entity. | |
| virtual Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const =0 |
| Evaluation of all reference shape functions in a number of points. | |
| virtual Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > | GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const =0 |
| Computation of the gradients of all reference shape functions in a number of points. | |
| virtual Eigen::MatrixXd | EvaluationNodes () const =0 |
| Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case. | |
| virtual size_type | NumEvaluationNodes () const =0 |
| Tell the number of evaluation (interpolation) nodes. | |
| virtual Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > | NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodvals) const |
| Computes the linear combination of reference shape functions matching function values at evaluation nodes. | |
Protected Member Functions | |
| ScalarReferenceFiniteElement ()=default | |
| ScalarReferenceFiniteElement (const ScalarReferenceFiniteElement &)=default | |
| ScalarReferenceFiniteElement (ScalarReferenceFiniteElement &&) noexcept=default | |
| ScalarReferenceFiniteElement & | operator= (const ScalarReferenceFiniteElement &)=default |
| ScalarReferenceFiniteElement & | operator= (ScalarReferenceFiniteElement &&) noexcept=default |
Related Symbols | |
(Note that these are not member symbols.) | |
| template<class SCALAR > | |
| void | PrintInfo (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &srfe, unsigned int ctrl=0) |
| Print information about a ScalarReferenceFiniteElement to the given stream. | |
| template<typename SCALAR > | |
| std::ostream & | operator<< (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &fe_desc) |
| Stream output operator: just calls the ScalarReferenceFiniteElement::print() method. | |
Detailed Description
class lf::fe::ScalarReferenceFiniteElement< SCALAR >
Interface class for parametric scalar valued finite elements.
- Template Parameters
-
SCALAR underlying scalar type, usually either doubleorcomplex<double>
A scalar parametric finite element is defined through a set of reference shape functions (RSFs) on a particular reference entity, cf. definition 2.8.3.1.
Each reference shape function is associated with a unique sub-entity of the reference entity according to Equation 2.8.1.6.
Specializations of this class support the evaluation of RSFs in arbitrary points in the reference element and the computation of their gradients. The also provide local components for the defintion of nodal interpolants.
This class is discussed in detail in Paragraph 2.8.3.25.
Numbering convention for reference shape functions
The numbering of reference shape functions is done according to the following convention, see also Paragraph 2.7.4.11
- RSFs belonging to sub-entities of a larger (relative) co-dimension are arranged before dofs associated with sub-entities of a smaller co-dimension (first nodes, then edges, finally cells).
- RSFs belonging to the same sub-entity are numbered contiguously.
- RSFs for sub-entities of the same co-dimension are taken into account in the order given by the local indexing of the sub-entities.
The numbering scheme must the same as that adopted for the definition of local-to-global maps, see the lf::assemble::DofHandler class.
Example for numbering of reference shape functions
For triangular cubic Lagrangian finite elements there is a single reference shape function associated with each vertex, two reference shape functions belonging to every edge and a single interior reference shape function. In this case the RSFs are numbered a follows
- RSF 0 -> vertex 0
- RSF 1 -> vertex 1
- RSF 2 -> vertex 2
- RSF 3,4 -> edge 0 (3 closer to endpoint 0, 4 closer to endpoint 1)
- RSF 5,6 -> edge 1 (5 closer to endpoint 0, 6 closer to endpoint 1)
- RSF 7,8 -> edge 2 (7 closer to endpoint 0, 8 closer to endpoint 1)
- RSF 9 -> triangle
The rules governing this numbering in LehrFEM++ are explained above and in Paragraph 2.7.4.11.
Definition at line 82 of file scalar_reference_finite_element.h.
Member Typedef Documentation
◆ Scalar
| using lf::fe::ScalarReferenceFiniteElement< SCALAR >::Scalar = SCALAR |
The underlying scalar type.
Definition at line 98 of file scalar_reference_finite_element.h.
Constructor & Destructor Documentation
◆ ScalarReferenceFiniteElement() [1/3]
|
protecteddefault |
◆ ScalarReferenceFiniteElement() [2/3]
|
protecteddefault |
◆ ScalarReferenceFiniteElement() [3/3]
|
protecteddefaultnoexcept |
◆ ~ScalarReferenceFiniteElement()
|
virtualdefault |
Virtual destructor
Member Function Documentation
◆ Degree()
|
pure virtual |
Request the maximal polynomial degree of the basis functions in this finite element.
Implemented in lf::fe::FePoint< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >.
Referenced by lf::fe::ScalarReferenceFiniteElement< SCALAR >::PrintInfo().
◆ Dimension()
|
inline |
Returns the spatial dimension of the reference cell.
Definition at line 119 of file scalar_reference_finite_element.h.
References lf::base::RefEl::Dimension(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::RefEl().
Referenced by lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions().
◆ EvalReferenceShapeFunctions()
|
pure virtual |
Evaluation of all reference shape functions in a number of points.
- Parameters
-
refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N, where dim is the dimension of the reference element, that is =0 for points, =1 for edges, =2 for triangles and quadrilaterals
- Returns
- An Eigen Matrix of size
NumRefShapeFunctions() x refcoords.cols()which contains the shape functions evaluated at every quadrature point.
Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler or the documentation of the class.
- Note
- shape functions are assumed to be real-valued.
Example: Triangular Linear Lagrangian finite elements
There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x2 matrix:
\[ \begin{pmatrix}\hat{b}^1(\mathbf{x}_1) & \hat{b}^1(\mathbf{x}_2) \\ \hat{b}^2(\mathbf{x}_1) & \hat{b}^2(\mathbf{x}_2) \\ \hat{b}^3(\mathbf{x}_1)\ & \hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]
Implemented in lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >, lf::fe::FePoint< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, and lf::uscalfe::FeLagrangeO3Quad< SCALAR >.
◆ EvaluationNodes()
|
pure virtual |
Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case.
- Returns
- A d x N matrix, where d is the dimension of the reference cell, and N the number of interpolation nodes. The columns of this matrix contain their reference coordinates.
Every parametric scalar finite element implicitly defines a local interpolation operator by duality with the reference shape functions. This interpolation operator can be realized through evaluations at certain evaluation nodes, which are provided by this method.
Unisolvence
The evaluation points must satisfy the following requirement: If the values of a function belonging to the span of the reference shape functions are known in the evaluation nodes, then this function is uniquely determined. This entails that the number of evaluation nodes must be at least as big as the number of reference shape functions.
- Note
- It is not required that any vector a values at evaluation nodes can be produced by a suitable linear combination of reference shape functions. For instance, this will not be possible, if there are more evaluation points than reference shape functions. If both sets have the same size, however, the interpolation problem has a solution for any vector of values at the evluation points.
Example: Principal lattice
For triangular Lagrangian finite elements of degree p the evaluation nodes, usually called "interpolation nodes" in this context, can be chosen as \(\left(\frac{j}{p},\frac{k}{p}\right),\; 0\leq j,k \leq p, j+k\leq p\).
Moment-based degrees of freedom
For some finite element spaces the interpolation functional may be defined based on integrals over edges. In this case the evaluation nodes will be quadrature nodes for the approximate evaluation of these integrals.
The quadrature rule must be exact for the polynomials contained in the local finite element spaces.
Implemented in lf::fe::FePoint< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >.
Referenced by lf::fe::ScalarReferenceFiniteElement< SCALAR >::PrintInfo().
◆ GradientsReferenceShapeFunctions()
|
pure virtual |
Computation of the gradients of all reference shape functions in a number of points.
- Parameters
-
refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N.
- Returns
- An Eigen Matrix of size
NumRefShapeFunctions() x (dim * refcoords.cols())wheredimis the dimension of the reference finite element. The gradients are returned in chunks of rows of this matrix.
Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler.
Example: Triangular Linear Lagrangian finite elements
There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x4 matrix:
\[ \begin{pmatrix} \grad^T\hat{b}^1(\mathbf{x}_1) & \grad^T\hat{b}^1(\mathbf{x}_2) \\ \grad^T\hat{b}^2(\mathbf{x}_1) & \grad^T\hat{b}^2(\mathbf{x}_2) \\ \grad^T\hat{b}^3(\mathbf{x}_1) & \grad^T\hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]
Implemented in lf::fe::FePoint< SCALAR >, lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, and lf::uscalfe::FeLagrangeO3Quad< SCALAR >.
◆ NodalValuesToDofs()
|
inlinevirtual |
Computes the linear combination of reference shape functions matching function values at evaluation nodes.
- Parameters
-
nodvals row vector of function values at evaluation nodes The length of this vector must agree with NumEvaluationNodes().
- Returns
- The coefficients of the local "nodal interpolant" with respect to the reference shape functions. This is a row vector of length NumRefShapeFunctions().
If the evaluation nodes are interpolation nodes, that is, if the set of reference shape functions forms a cardinal basis with respect to these nodes, then we have NumEvaluationNodes() == NumRefShapeFunctions() and the linear mapping realized by NodalValuesToDofs() is the identity mapping.
- Note
- default implementation is the identity mapping
Requirement: reproduction of local finite element functions
If the vector of values at the evaluation nodes agrees with a vector of function values of a linear combination of reference shape functions at the evaluation nodes, then this method must return the very coefficients of the linear combination.
Reimplemented in lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >.
Definition at line 311 of file scalar_reference_finite_element.h.
References lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumEvaluationNodes().
◆ NumEvaluationNodes()
|
pure virtual |
Tell the number of evaluation (interpolation) nodes.
Implemented in lf::fe::FePoint< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >.
Referenced by lf::fe::ScalarReferenceFiniteElement< SCALAR >::NodalValuesToDofs(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::PrintInfo().
◆ NumRefShapeFunctions() [1/3]
|
inlinevirtual |
Total number of reference shape functions associated with the reference cell.
- Note
- the total number of shape functions is the sum of the number of interior shape functions for all sub-entities and the entity itself.
Reimplemented in lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >.
Definition at line 128 of file scalar_reference_finite_element.h.
References lf::fe::ScalarReferenceFiniteElement< SCALAR >::Dimension(), lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions(), lf::base::RefEl::NumSubEntities(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::RefEl().
Referenced by lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::PrintInfo().
◆ NumRefShapeFunctions() [2/3]
|
inlinevirtual |
The number of interior reference shape functions for sub-entities of a particular co-dimension.
- Parameters
-
codim co-dimension of the subentity
- Returns
- number of interior reference shape function belonging to the specified sub-entity.
- Note
- this method will throw an exception when different numbers of reference shape functions are assigned to different sub-entities of the same co-dimension
Reimplemented in lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >, and lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >.
Definition at line 152 of file scalar_reference_finite_element.h.
◆ NumRefShapeFunctions() [3/3]
|
pure virtual |
The number of interior reference shape functions for every sub-entity.
- Parameters
-
codim do-dimension of the subentity subidx local index of the sub-entity
- Returns
- number of interior reference shape function belonging to the specified sub-entity.
Implemented in lf::fe::FePoint< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, and lf::uscalfe::FeLagrangeO2Quad< SCALAR >.
◆ operator=() [1/2]
|
protecteddefault |
◆ operator=() [2/2]
|
protecteddefaultnoexcept |
◆ RefEl()
|
pure virtual |
Tells the type of reference cell underlying the parametric finite element.
Implemented in lf::fe::FePoint< SCALAR >, lf::fe::FeHierarchicSegment< SCALAR >, lf::fe::FeHierarchicTria< SCALAR >, lf::fe::FeHierarchicQuad< SCALAR >, lf::fe::test_utils::ComplexScalarReferenceFiniteElement< SCALAR >, lf::uscalfe::FeLagrangeO1Tria< SCALAR >, lf::uscalfe::FeLagrangeO1Quad< SCALAR >, lf::uscalfe::FeLagrangeO1Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Tria< SCALAR >, lf::uscalfe::FeLagrangeO2Segment< SCALAR >, lf::uscalfe::FeLagrangeO2Quad< SCALAR >, lf::uscalfe::FeLagrangeO3Tria< SCALAR >, lf::uscalfe::FeLagrangeO3Segment< SCALAR >, lf::uscalfe::FeLagrangeO3Quad< SCALAR >, and lf::uscalfe::PrecomputedScalarReferenceFiniteElement< SCALAR >.
Referenced by lf::fe::ScalarReferenceFiniteElement< SCALAR >::Dimension(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions().
Friends And Related Symbol Documentation
◆ operator<<()
|
related |
Stream output operator: just calls the ScalarReferenceFiniteElement::print() method.
Definition at line 353 of file scalar_reference_finite_element.h.
◆ PrintInfo()
|
related |
Print information about a ScalarReferenceFiniteElement to the given stream.
- Parameters
-
o stream to which output is to be sent srfe The ScalarReferenceFiniteElement that should be printed. ctrl controls the level of detail that is printed (see below)
Level of output:
- ctrl = 0: Only the type of the FiniteElementSpace, degree and number of ShapeFunctions/Evaluation nodes are printed
- ctrl > 0: Also the coordinates of the evaluation nodes are printed.
Definition at line 337 of file scalar_reference_finite_element.h.
References lf::fe::ScalarReferenceFiniteElement< SCALAR >::Degree(), lf::fe::ScalarReferenceFiniteElement< SCALAR >::EvaluationNodes(), lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumEvaluationNodes(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::NumRefShapeFunctions().
The documentation for this class was generated from the following file:
- /__w/lehrfempp/lehrfempp/lib/lf/fe/scalar_reference_finite_element.h
