FEValues

CellScalarValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])
CellVectorValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])

A CellValues object facilitates the process of evaluating values of shape functions, gradients of shape functions, values of nodal functions, gradients and divergences of nodal functions etc. in the finite element cell. There are two different types of CellValues: CellScalarValues and CellVectorValues. As the names suggest, CellScalarValues utilizes scalar shape functions and CellVectorValues utilizes vectorial shape functions. For a scalar field, the CellScalarValues type should be used. For vector field, both subtypes can be used.

Arguments:

• T: an optional argument (default to Float64) to determine the type the internal data is stored as.
• quad_rule: an instance of a QuadratureRule
• func_interpol: an instance of an Interpolation used to interpolate the approximated function
• geom_interpol: an optional instance of a Interpolation which is used to interpolate the geometry

Common methods:

• reinit!

• getnquadpoints

• getdetJdV

• shape_value

• shape_gradient

• shape_symmetric_gradient

• shape_divergence

• function_value

• function_gradient

• function_symmetric_gradient

• function_divergence

• spatial_coordinate

reinit!(cv::CellValues, x::Vector)
reinit!(bv::FaceValues, x::Vector, face::Int)

Update the CellValues/FaceValues object for a cell or face with coordinates x. The derivatives of the shape functions, and the new integration weights are computed.

getnquadpoints(fe_v::Values)

Return the number of quadrature points for the Values object.

getdetJdV(fe_v::Values, q_point::Int)

Return the product between the determinant of the Jacobian and the quadrature point weight for the given quadrature point: $\det(J(\mathbf{x})) w_q$

This value is typically used when one integrates a function on a finite element cell or face as

$\int\limits_\Omega f(\mathbf{x}) d \Omega \approx \sum\limits_{q = 1}^{n_q} f(\mathbf{x}_q) \det(J(\mathbf{x})) w_q$ $\int\limits_\Gamma f(\mathbf{x}) d \Gamma \approx \sum\limits_{q = 1}^{n_q} f(\mathbf{x}_q) \det(J(\mathbf{x})) w_q$

shape_value(fe_v::Values, q_point::Int, base_function::Int)

Return the value of shape function base_function evaluated in quadrature point q_point.

shape_gradient(fe_v::Values, q_point::Int, base_function::Int)

Return the gradient of shape function base_function evaluated in quadrature point q_point.

shape_symmetric_gradient(fe_v::Values, q_point::Int, base_function::Int)

Return the symmetric gradient of shape function base_function evaluated in quadrature point q_point.

shape_divergence(fe_v::Values, q_point::Int, base_function::Int)

Return the divergence of shape function base_function evaluated in quadrature point q_point.

function_value(fe_v::Values, q_point::Int, u::AbstractVector)

Compute the value of the function in a quadrature point. u is a vector with values for the degrees of freedom. For a scalar valued function, u contains scalars. For a vector valued function, u can be a vector of scalars (for use of VectorValues) or u can be a vector of Vecs (for use with ScalarValues).

The value of a scalar valued function is computed as $u(\mathbf{x}) = \sum\limits_{i = 1}^n N_i (\mathbf{x}) u_i$ where $u_i$ are the value of $u$ in the nodes. For a vector valued function the value is calculated as $\mathbf{u}(\mathbf{x}) = \sum\limits_{i = 1}^n N_i (\mathbf{x}) \mathbf{u}_i$ where $\mathbf{u}_i$ are the nodal values of $\mathbf{u}$.

function_gradient(fe_v::Values{dim}, q_point::Int, u::AbstractVector)

Compute the gradient of the function in a quadrature point. u is a vector with values for the degrees of freedom. For a scalar valued function, u contains scalars. For a vector valued function, u can be a vector of scalars (for use of VectorValues) or u can be a vector of Vecs (for use with ScalarValues).

The gradient of a scalar function or a vector valued function with use of VectorValues is computed as $\mathbf{\nabla} u(\mathbf{x}) = \sum\limits_{i = 1}^n \mathbf{\nabla} N_i (\mathbf{x}) u_i$ or $\mathbf{\nabla} u(\mathbf{x}) = \sum\limits_{i = 1}^n \mathbf{\nabla} \mathbf{N}_i (\mathbf{x}) u_i$ respectively, where $u_i$ are the nodal values of the function. For a vector valued function with use of ScalarValues the gradient is computed as $\mathbf{\nabla} \mathbf{u}(\mathbf{x}) = \sum\limits_{i = 1}^n \mathbf{u}_i \otimes \mathbf{\nabla} N_i (\mathbf{x})$ where $\mathbf{u}_i$ are the nodal values of $\mathbf{u}$.

function_symmetric_gradient(fe_v::Values, q_point::Int, u::AbstractVector)

Compute the symmetric gradient of the function, see function_gradient. Return a SymmetricTensor.

The symmetric gradient of a scalar function is computed as $\left[ \mathbf{\nabla} \mathbf{u}(\mathbf{x_q}) \right]^\text{sym} = \sum\limits_{i = 1}^n \frac{1}{2} \left[ \mathbf{\nabla} N_i (\mathbf{x}_q) \otimes \mathbf{u}_i + \mathbf{u}_i \otimes \mathbf{\nabla} N_i (\mathbf{x}_q) \right]$ where $\mathbf{u}_i$ are the nodal values of the function.

function_divergence(fe_v::Values, q_point::Int, u::AbstractVector)

Compute the divergence of the vector valued function in a quadrature point.

The divergence of a vector valued functions in the quadrature point $\mathbf{x}_q)$ is computed as $\mathbf{\nabla} \cdot \mathbf{u}(\mathbf{x_q}) = \sum\limits_{i = 1}^n \mathbf{\nabla} N_i (\mathbf{x_q}) \cdot \mathbf{u}_i$ where $\mathbf{u}_i$ are the nodal values of the function.

spatial_coordinate(fe_v::Values{dim}, q_point::Int, x::AbstractVector)

Compute the spatial coordinate in a quadrature point. x contains the nodal coordinates of the cell.

The coordinate is computed, using the geometric interpolation, as $\mathbf{x} = \sum\limits_{i = 1}^n M_i (\mathbf{x}) \mathbf{\hat{x}}_i$

FaceScalarValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])
FaceVectorValues([::Type{T}], quad_rule::QuadratureRule, func_interpol::Interpolation, [geom_interpol::Interpolation])

A FaceValues object facilitates the process of evaluating values of shape functions, gradients of shape functions, values of nodal functions, gradients and divergences of nodal functions etc. on the faces of finite elements. There are two different types of FaceValues: FaceScalarValues and FaceVectorValues. As the names suggest, FaceScalarValues utilizes scalar shape functions and FaceVectorValues utilizes vectorial shape functions. For a scalar field, the FaceScalarValues type should be used. For vector field, both subtypes can be used.

Arguments:

• T: an optional argument to determine the type the internal data is stored as.
• quad_rule: an instance of a QuadratureRule
• func_interpol: an instance of an Interpolation used to interpolate the approximated function
• geom_interpol: an optional instance of an Interpolation which is used to interpolate the geometry

Common methods:

• reinit!

• getnquadpoints

• getdetJdV

• shape_value

• shape_gradient

• shape_symmetric_gradient

• shape_divergence

• function_value

• function_gradient

• function_symmetric_gradient

• function_divergence

• spatial_coordinate

getcurrentface(fv::FaceValues)

Return the current active face of the FaceValues object (from last reinit!).