Interpolations

Interpolation{refshape, order}()

Abstract type for interpolations defined on refshape (see AbstractRefShape). order corresponds to the order of the interpolation. The interpolation is used to define shape functions to interpolate a function between nodes.

The following interpolations are implemented:

• Lagrange
• DiscontinuousLagrange
• Serendipity
• BubbleEnrichedLagrange
• CrouzeixRaviart
• RannacherTurek
• RaviartThomas
• BrezziDouglasMarini
• Nedelec

See the docstring of each interpolation for the supported reference shapes and orders. Scalar interpolations can be vectorized, i.e. used for each component of a vector field, see VectorizedInterpolation.

Examples

julia> ip = Lagrange{RefTriangle, 2}()Lagrange{RefTriangle, 2}()julia> getnbasefunctions(ip)6

getnbasefunctions(ip::Interpolation)

Return the number of base functions for the interpolation ip.

Ferrite.getrefdim(::Interpolation)

Return the dimension of the reference element for a given interpolation.

Ferrite.getrefshape(::Interpolation)::AbstractRefShape

Return the reference element shape of the interpolation.

Ferrite.getorder(::Interpolation)

Return order of the interpolation.

Lagrange{refshape, order} <: ScalarInterpolation

Standard continuous Lagrange polynomials with equidistant node placement.

See also the Lagrange element on DefElement.

DiscontinuousLagrange{refshape, order} <: ScalarInterpolation

Piecewise discontinuous Lagrange basis via Gauss-Lobatto points.

See also the discontinuous Lagrange element on DefElement.

Serendipity{refshape, order} <: ScalarInterpolation

Serendipity element on hypercubes. Currently only second order variants are implemented.

See also the serendipity element on DefElement.

BubbleEnrichedLagrange{refshape, order} <: ScalarInterpolation

Lagrange element with bubble stabilization.

Currently only BubbleEnrichedLagrange{RefTriangle, 1} is implemented.

See also the bubble enriched Lagrange element on DefElement.

CrouzeixRaviart{refshape, order} <: ScalarInterpolation

Classical non-conforming Crouzeix–Raviart element.

For details we refer to the original paper [13].

See also the Crouzeix–Raviart element on DefElement.

RannacherTurek{refshape, order} <: ScalarInterpolation

Classical non-conforming Rannacher-Turek element.

This element is basically the idea from Crouzeix and Raviart applied to hypercubes. For details see the original paper [14].

See also the Rannacher–Turek element on DefElement.

VectorizedInterpolation{vdim}(ip::ScalarInterpolation)
VectorizedInterpolation(ip::ScalarInterpolation)
ip^vdim

Vector valued interpolation with vdim components where each component is interpolated using the scalar interpolation ip. If vdim is not given it defaults to the reference dimension of ip. A VectorizedInterpolation is typically constructed using the ^ syntax, e.g. Lagrange{RefTriangle, 2}()^2.

Examples

julia> ip = Lagrange{RefTriangle, 2}()^2Lagrange{RefTriangle, 2}()^2julia> getnbasefunctions(ip)12

RaviartThomas{refshape, order} <: VectorInterpolation

Raviart-Thomas element (of the first kind) for $H(\mathrm{div})$-conforming discretizations, i.e. the normal component is continuous across cell boundaries.

The following combinations of reference shape and order are implemented: RefTriangle (order 1 and 2), RefQuadrilateral (order 1), RefTetrahedron (order 1), and RefHexahedron (order 1).

See also the Raviart–Thomas element on DefElement.

BrezziDouglasMarini{refshape, order} <: VectorInterpolation

Brezzi-Douglas-Marini element for $H(\mathrm{div})$-conforming discretizations, i.e. the normal component is continuous across cell boundaries.

Currently only BrezziDouglasMarini{RefTriangle, 1} is implemented.

See also the Brezzi–Douglas–Marini element on DefElement.

Nedelec{refshape, order} <: VectorInterpolation

Nédélec element (of the first kind) for $H(\mathrm{curl})$-conforming discretizations, i.e. the tangential component is continuous across cell boundaries.

The following combinations of reference shape and order are implemented: RefTriangle (order 1 and 2), RefQuadrilateral (order 1), RefTetrahedron (order 1), and RefHexahedron (order 1).

See also the Nédélec (first kind) element on DefElement.