\(k\)-form Type

While Degrees of Freedom may be used to describe a function defined on a reference domain, this is not very useful when dealing with \(k\)-forms, since they have multiple components, with their functions spaces being variations of the same base space, depending on their covector basis bundle. As such, two types are provided:

• To specify and obtain information about \(k\)-forms there is KFormSpecs type.

• To hold and manipulate degrees of freedom of components there is KForm type.

class fdg.KFormSpecs(order: int, base_space: FunctionSpace)

Differential k-form specification.

Parameters:

• order (int) – Order of the k-form.

• base_space (FunctionSpace) – Base space to use for the k-forms. This is also the space in which 0-forms are defined.

get_component_basis(idx: int) → CovectorBasis

Get covector basis bundle for a component.

Parameters:

idx (int) – Index of the component.

Returns:

Covector basis bundle corresponding to the k-form component with the specified index.

Return type:

CovectorBasis

get_component_function_space(idx: int) → FunctionSpace

Get the function space for a component.

Parameters:

idx (int) – Index of the component.

Returns:

Function space corresponding to the k-form component with the specified index.

Return type:

FunctionSpace

get_component_slice(idx: int) → slice

Get the slice corresponding to degrees of freedom of a k-form component.

The resulting slice can be used to index into the flattened array of degrees of freedom to get the DoFs corresponding to a praticular component.

Parameters:

idx (int) – Index of the k-form component.

Returns:

Slice of the flattened array of all k-form degrees of freedom that corresponds to degrees of freedom of the specified component.

Return type:

slice

base_space

Base function space the k-form is based in.

Type:

FunctionSpace

component_count

Number of components in the k-form.

Type:

int

component_dof_counts

Number of DoFs in each component.

Type:

numpy.typing.NDArray[numpy.int64]

dimension

Dimension of the space the k-form is in.

Type:

int

order

Order of the k-form.

Type:

int

class fdg.KForm(specs: KFormSpecs)

Type holding the degrees of freedom of a k-form.

Parameters:

specs (KFormSpecs) – Specification of the k-form that is to be created.

get_component_dofs(idx: int) → numpy.tying.NDArray[numpy.double]

Get the array containing the degrees of freedom for a k-form component.

Parameters:

idx (int) – Index of the k-form component.

Returns:

Array containing the degrees of freedom. This is not a copy, so changing values in it will change the values of degrees of freedom.

Return type:

array

specs

Specifications of the k-form.

Type:

KFormSpecs

values

Values of all k-form degrees of freedom.

Type:

numpy.typing.NDArray[numpy.double]