:py:mod:`symfem.elements.dual`
==============================

.. py:module:: symfem.elements.dual

.. autoapi-nested-parse::

   Dual elements.

   These elements' definitions appear in https://doi.org/10.1016/j.crma.2004.12.022
   (Buffa, Christiansen, 2005)



Module Contents
---------------

Classes
~~~~~~~

.. autoapisummary::

   symfem.elements.dual.DualCiarletElement
   symfem.elements.dual.Dual
   symfem.elements.dual.BuffaChristiansen
   symfem.elements.dual.RotatedBuffaChristiansen




.. py:class:: DualCiarletElement(dual_coefficients: List[List[List[Union[int, sympy.core.expr.Expr]]]], fine_space: str, reference: symfem.references.DualPolygon, order: int, dof_entities: List[Tuple[int, int]], domain_dim: int, range_dim: int, range_shape: Optional[Tuple[int, Ellipsis]] = None, dof_directions: Optional[symfem.geometry.SetOfPoints] = None)


   Bases: :py:obj:`symfem.finite_element.FiniteElement`

   Abstract barycentric finite element.

   .. py:property:: maximum_degree
      :type: int
      :abstractmethod:

      Get the maximum degree of this polynomial set for the element.

   .. py:method:: get_polynomial_basis(reshape: bool = True) -> List[symfem.functions.AnyFunction]

      Get the symbolic polynomial basis for the element.

      :returns: The polynomial basis


   .. py:method:: get_dual_matrix() -> sympy.matrices.dense.MutableDenseMatrix

      Get the dual matrix.

      :returns: The dual matrix


   .. py:method:: get_basis_functions(use_tensor_factorisation: bool = False) -> List[symfem.functions.AnyFunction]

      Get the basis functions of the element.

      :param use_tensor_factorisation: Should a tensor factorisation be used?

      :returns: The basis functions


   .. py:method:: entity_dofs(entity_dim: int, entity_number: int) -> List[int]

      Get the numbers of the DOFs associated with the given entity.

      :param entity_dim: The dimension of the entity
      :param entity_number: The number of the entity

      :returns: The numbers of the DOFs associated with the entity


   .. py:method:: dof_plot_positions() -> List[symfem.geometry.PointType]

      Get the points to plot each DOF at on a DOF diagram.

      :returns: The DOF positions


   .. py:method:: dof_directions() -> List[Union[symfem.geometry.PointType, None]]

      Get the direction associated with each DOF.

      :returns: The DOF directions


   .. py:method:: dof_entities() -> List[Tuple[int, int]]

      Get the entities that each DOF is associated with.

      :returns: The entities


   .. py:method:: map_to_cell(vertices_in: symfem.geometry.SetOfPointsInput, basis: Optional[List[symfem.functions.AnyFunction]] = None, forward_map: Optional[symfem.geometry.PointType] = None, inverse_map: Optional[symfem.geometry.PointType] = None) -> List[symfem.functions.AnyFunction]
      :abstractmethod:

      Map the basis onto a cell using the appropriate mapping for the element.

      :param vertices_in: The vertices of the cell
      :param basis: The basis functions
      :param forward_map: The map from the reference to the cell
      :param inverse_map: The map to the reference from the cell

      :returns: The basis functions mapped to the cell



.. py:class:: Dual(reference: symfem.references.DualPolygon, order: int)


   Bases: :py:obj:`DualCiarletElement`

   Barycentric dual finite element.

   .. py:attribute:: names
      :value: ['dual polynomial', 'dual P', 'dual']

      

   .. py:attribute:: references
      :value: ['dual polygon']

      

   .. py:attribute:: min_order
      :value: 0

      

   .. py:attribute:: max_order
      :value: 1

      

   .. py:attribute:: continuity
      :value: 'C0'

      

   .. py:attribute:: last_updated
      :value: '2023.05'

      


.. py:class:: BuffaChristiansen(reference: symfem.references.DualPolygon, order: int)


   Bases: :py:obj:`DualCiarletElement`

   Buffa-Christiansen barycentric dual finite element.

   .. py:attribute:: names
      :value: ['Buffa-Christiansen', 'BC']

      

   .. py:attribute:: references
      :value: ['dual polygon']

      

   .. py:attribute:: min_order
      :value: 1

      

   .. py:attribute:: max_order
      :value: 1

      

   .. py:attribute:: continuity
      :value: 'H(div)'

      

   .. py:attribute:: last_updated
      :value: '2023.05'

      


.. py:class:: RotatedBuffaChristiansen(reference: symfem.references.DualPolygon, order: int)


   Bases: :py:obj:`DualCiarletElement`

   RotatedBuffa-Christiansen barycentric dual finite element.

   .. py:attribute:: names
      :value: ['rotated Buffa-Christiansen', 'RBC']

      

   .. py:attribute:: references
      :value: ['dual polygon']

      

   .. py:attribute:: min_order
      :value: 1

      

   .. py:attribute:: max_order
      :value: 1

      

   .. py:attribute:: continuity
      :value: 'H(div)'

      

   .. py:attribute:: last_updated
      :value: '2023.05'