Let us recall that all finite element methods defined in *GetFEM++* are declared in the
file `getfem_fem.h` and that a descriptor on a finite element method is obtained
thanks to the function:

```
getfem::pfem pf = getfem::fem_descriptor("name of method");
```

where `"name of method"` is a string to be choosen among the existing methods.

It is possible to define a classical Lagrange element of arbitrary
dimension and arbitrary degree. Each degree of freedom of such an element
corresponds to the value of the function on a corresponding node. The grid of
node is the so-called Lagrange grid. Figures *Examples of classical Lagrange elements on a segment*.

The number of degrees of freedom for a classical Lagrange element of dimension and degree is . For instance, in dimension 2 , this value is and in dimension 3 , it is .

The particular way used in *GetFEM++* to numerate the nodes are also shown in figures
*segment*, *triangle* and
*tetrahedron*. Using another numeration, let

be somme indices such that

Then, the coordinate of a node can be computed as

where are the vertices of the simplex (for the particular choice has been chosen). Then each base function, corresponding of each node is defined by

where are the barycentric coordinates, i.e. the polynomials of degree 1 whose value is on the vertex and whose value is on other vertices. On the reference element, one has

When between two elements of the same degrees (even with different dimensions),
the d.o.f. of a common face are linked, the element is of class . This
means that the global polynomial is continuous. If you try to link elements of
different degrees, you will get some trouble with the unlinked d.o.f. This is not
automatically supported by *GetFEM++*, so you will have to support it (add constraints
on these d.o.f.).

For some applications (computation of a gradient for instance) one may not want the d.o.f. of a common face to be linked. This is why there are two versions of the classical Lagrange element.

Classical Lagrange element "FEM_PK(P, K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

Discontinuous Lagrange element "FEM_PK_DISCONTINUOUS(P, K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , discontinuous No Yes Yes

Even though Lagrange elements are defined for arbitrary degrees, to choose a high degree can be problematic for a large number of applications due to the “noisy” caracteristic of the lagrange basis. These elements are recommended for the basic interpolation but for p.d.e. applications elements with hierarchical basis are preferable (see the corresponding section).

Classical Lagrange elements on parallelepipeds or prisms are obtained as tensor product of Lagrange elements on simplices. When two elements are defined, one on a dimension and the other in dimension , one obtains the base functions of the tensorial product (on the reference element) as

where and are respectively the base functions of the first and second element.

The element on a parallelepiped of dimension is obtained as
the tensorial product of classical elements on the segment.
Examples in dimension 2 are shown in figure *dimension 2*
and in dimension 3 in figure *dimension 3*.

A prism in dimension is the direct product of a simplex of dimension
with a segment. The element on this prism is
the tensorial product of the classical element on a simplex of
dimension with the classical element on a segment. For
this coincide with a parallelepiped. Examples in dimension
are shown in figure *dimension 3*. This is also possible
not to have the same degree on each dimension. An example is shown on figure
*dimension 3, prism*.

. Lagrange element on parallelepipeds "FEM_QK(P, K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

. Lagrange element on prisms "FEM_PK_PRISM(P, K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

. Lagrange element on prisms "FEM_PRODUCT(FEM_PK(P-1, K1), FEM_PK(1, K2))"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

Incomplete Lagrange element on parallelepipeds (Quad 8 and Hexa 20 serendipity elements) "FEM_Q2_INCOMPLETE(P)"degree dimension d.o.f. number class vector -equivalent Polynomial 3 , No Yes Yes

The idea behind hierarchical basis is the description of the solution at different level: a rough level, a more refined level ... In the same discretization some degrees of freedom represent the rough description, some other the more rafined and so on. This corresponds to imbricated spaces of discretization. The hierarchical basis contains a basis of each of these spaces (this is not the case in classical Lagrange elements when the mesh is refined).

Among the advantages, the condition number of rigidity matrices can be greatly improved, it allows local raffinement and a resolution with a multigrid approach.

. Classical Lagrange element on simplices but with a hierarchical basis with respect to the degree "FEM_PK_HIERARCHICAL(P,K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

. Classical Lagrange element on parallelepipeds but with a hierarchical basis with respect to the degree "FEM_QK_HIERARCHICAL(P,K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

. Classical Lagrange element on prisms but with a hierarchical basis with respect to the degree "FEM_PK_PRISM_HIERARCHICAL(P,K)"degree dimension d.o.f. number class vector -equivalent Polynomial , , No Yes Yes

some particular choices: will be built with the basis of the , the additional basis of the then the additional basis of the .

will be built with the basis of the , the additional basis
:of the then the additional basis of the (not with the
:basis of the , the additional basis of the then the
:additional basis of the , it is possible to build the latter with
:`"FEM_GEN_HIERARCHICAL(a,b)"`)

The principal interest of the composite elements is to build hierarchical elements. But this tool can also be used to build piecewise polynomial elements.

Composition of a finite element method on an element with Ssubdivisions"FEM_STRUCTURED_COMPOSITE(FEM1, S)"degree dimension d.o.f. number class vector -equivalent Polynomial degree of FEM1 dimension of FEM1 variable variable No If FEM1ispiecewise

It is important to use a corresponding composite integration method.

Hierarchical composition of a finite element method on a simplex with Ssubdivisions"FEM_PK_HIERARCHICAL_COMPOSITE(P,K,S)"degree dimension d.o.f. number class vector -equivalent Polynomial variable No Yes piecewise

Hierarchical composition of a hierarchical finite element method on a simplex with Ssubdivisions"FEM_PK_FULL_HIERARCHICAL_COMPOSITE(P,K,S)"degree dimension d.o.f. number class vector -equivalent Polynomial variable No Yes piecewise

Other constructions are possible thanks to `"FEM_GEN_HIERARCHICAL(FEM1, FEM2)"`
and `"FEM_STRUCTURED_COMPOSITE(FEM1, S)"`.

It is important to use a corresponding composite integration method.

Raviart-Thomas of lowest order element on simplices "FEM_RT0(P)"degree dimension d.o.f. number class vector -equivalent Polynomial H(div) Yes No Yes

Raviart-Thomas of lowest order element on parallelepipeds (quadrilaterals, hexahedrals) "FEM_RT0Q(P)"degree dimension d.o.f. number class vector -equivalent Polynomial H(div) Yes No Yes

Nedelec (or Whitney) edge element “FEM_NEDELEC(P)”` degree dimension d.o.f. number class vector -equivalent Polynomial H(rot) Yes No Yes

The 1D GaussLobatto element is similar to the classical fem on the segment, but the nodes are given by the Gauss-Lobatto-Legendre quadrature rule of order . This FEM is known to lead to better conditioned linear systems, and can be used with the corresponding quadrature to perform mass-lumping (on segments or parallelepipeds).

The polynomials coefficients have been pre-computed with Maple (they require the inversion of an ill-conditioned system), hence they are only available for the following values of : . Note that for and , this is the classical and fem.

GaussLobatto element on the segment "FEM_PK_GAUSSLOBATTO1D(K)"degree dimension d.o.f. number class vector -equivalent Polynomial No Yes Yes

Base functions on the reference element

This element is close to be -equivalent but it is not. On the real element the value of the gradient on vertices will be multiplied by the gradient of the geometric transformation. The matrix is not equal to identity but is still diagonal.

Hermite element on the segment "FEM_HERMITE(1)"degree dimension d.o.f. number class vector -equivalent Polynomial No No Yes

Lagrange element with an additional internal bubble function "FEM_PK_WITH_CUBIC_BUBBLE(1, 1)"degree dimension d.o.f. number class vector -equivalent Polynomial No Yes Yes

Lagrange or element with an additional internal bubble function "FEM_PK_WITH_CUBIC_BUBBLE(2, K)"degree dimension d.o.f. number class vector -equivalent Polynomial or No Yes Yes

Lagrange with an additional internal piecewise linear bubble function "FEM_P1_PIECEWISE_LINEAR_BUBBLE"degree dimension d.o.f. number class vector -equivalent Polynomial or No Yes Piecewise

Lagrange element with an additional bubble function on face 0 "FEM_P1_BUBBLE_FACE(2)"degree dimension d.o.f. number class vector -equivalent Polynomial No Yes Yes

. Lagrange element on a triangle with additional d.o.f on face 0 "FEM_P1_BUBBLE_FACE_LAG"degree dimension d.o.f. number class vector -equivalent Polynomial No Yes Yes

. non-conforming element on a triangle "FEM_P1_NONCONFORMING"degree dimension d.o.f. number class vector -equivalent Polynomial No Yes Yes

Base functions on the reference element:

This element is not -equivalent (The matrix is not equal to identity). On the real element linear combinations of and are used to match the gradient on the corresponding vertex. Idem for the two couples , and , for the two other vertices.

Hermite element on a triangle "FEM_HERMITE(2)"degree dimension d.o.f. number class vector -equivalent Polynomial No No Yes

This element is not -equivalent (The matrix is not equal to identity). In particular, it can be used for non-conforming discretization of fourth order problems, despite the fact that it is not .

Morley element on a triangle "FEM_MORLEY"degree dimension d.o.f. number class vector -equivalent Polynomial discontinuous No No Yes

The base functions on the reference element are:

This element is not -equivalent (The matrix is not equal to identity). On the real element linear combinations of the transformed base functions are used to match the gradient, the second derivatives and the normal derivatives on the faces. Note that the use of the matrix allows to define Argyris element even with nonlinear geometric transformations (for instance to treat curved boundaries).

Argyris element on a triangle "FEM_ARGYRIS"degree dimension d.o.f. number class vector -equivalent Polynomial No No Yes

This element is not -equivalent. This is a composite element.
Polynomial of degree 3 on each of the three sub-triangles (see figure
*Hsieh-Clough-Tocher (HCT) element, , 12 d.o.f., * and [ciarlet1978]). It is strongly advised to use a
`"IM_HCT_COMPOSITE"` integration method with this finite element. The numeration
of the dof is the following: 0, 3 and 6 for the lagrange dof on the first second
and third vertex respectively; 1, 4, 7 for the derivative with respects to the
first variable; 2, 5, 8 for the derivative with respects to the second variable
and 9, 10, 11 for the normal derivatives on face 0, 1, 2 respectively.

HCT element on a triangle "FEM_HCT_TRIANGLE"degree dimension d.o.f. number class vector -equivalent Polynomial No No piecewise

This element exists also in its reduced form, where the normal derivatives are
assumed to be polynomial of degree one on each edge (see figure
*Reduced Hsieh-Clough-Tocher (reduced HCT) element, , 9 d.o.f., *)

Reduced HCT element on a triangle "FEM_REDUCED_HCT_TRIANGLE"degree dimension d.o.f. number class vector -equivalent Polynomial No No piecewise

This element is not -equivalent. This is a composite element.
Polynomial of degree 3 on each of the four sub-triangles (see figure
*Composite element on quadrilaterals, piecewise , 16 d.o.f., *). At least on the reference element it corresponds to the
Fraeijs de Veubeke-Sander element (see [ciarlet1978]). It is strongly advised
to use a `"IM_QUADC1_COMPOSITE"` integration method with this finite element.

. composite element on a quadrilateral (FVS) "FEM_QUADC1_COMPOSITE"degree dimension d.o.f. number class vector -equivalent Polynomial No No piecewise

This element exists also in its reduced form, where the normal derivatives are
assumed to be polynomial of degree one on each edge (see figure
*Reduced composite element on quadrilaterals, piecewise , 12 d.o.f., *)

Reduced composite element on a quadrilateral (reduced FVS) "FEM_REDUCED_QUADC1_COMPOSITE"degree dimension d.o.f. number class vector -equivalent Polynomial No No piecewise

*GetFEM++* proposes some Lagrange pyramidal elements of degree 0, 1 and two based on [GR-GH1999] and [BE-CO-DU2010]. See these references for more details. The proposed element can be raccorded to standard or Lagrange fem on the triangular faces and to a standard or Lagrange fem on the quatrilateral face.

Degree 0 pyramidal element with 1 dof | Degree 1 pyramidal element with 5 dof | Degree 2 pyramidal element with 14 dof |

The associated geometric transformations are `"GT_PYRAMID(K)"` for K = 1 or 2. The associated integration methods `"IM_PYRAMID(im)"` where `im` is an integration method on a hexahedron (or alternatively `"IM_PYRAMID_COMPOSITE(im)"` where `im` is an integration method on a tetrahedron, but it is theoretically less accurate)
The shape functions are not polynomial ones but rational fractions. For the first degree the shape functions read:

For the second degree, setting

the shape functions read:

degree | dimension | d.o.f. number | class | vector | -equivalent | Polynomial |
---|---|---|---|---|---|---|

discontinuous | No | Yes | No | |||

No | Yes | No | ||||

No | Yes | No |

degree | dimension | d.o.f. number | class | vector | -equivalent | Polynomial |
---|---|---|---|---|---|---|

discontinuous | No | Yes | No | |||

discontinuous | No | Yes | No | |||

discontinuous | No | Yes | No |

Lagrange element with an additional internal bubble function "FEM_PK_WITH_CUBIC_BUBBLE(3, K)"degree dimension d.o.f. number class vector -equivalent Polynomial , or No Yes Yes

Lagrange element with an additional bubble function on face 0 "FEM_P1_BUBBLE_FACE(3)"degree dimension d.o.f. number class vector -equivalent Polynomial No Yes Yes

Base functions on the reference element:

This element is not -equivalent (The matrix is not equal to identity). On the real element linear combinations of , and are used to match the gradient on the corresponding vertex. Idem on the other vertices.

Hermite element on a tetrahedron "FEM_HERMITE(3)"degree dimension d.o.f. number class vector -equivalent Polynomial No No Yes