# Discretisation

For the discretisation of an equation, one has basically to define a Finite Element space - or more precisely, a tensor product of finite element spaces. These mathematical structures are realised in FEAT2 by two "discretisation" structures:

1. t_scalarDiscretisation (spatialdiscretisation.f90)

• Encapsules a finite element space $V_h$ over a mesh, e.g., the $Q_1$ space.
• Is connected to a mesh, (possibly) a domain description and transformation rules from reference to real elements.

2. t_blockDiscretisation (spatialdiscretisation.f90)

• Encapsules tensor product spaces of finite element spaces, e.g., $X_h = V_h \times V_h$.
• Is realised as a list of t_scalarDiscretisation structures, each describing a finite element subspace in $X_h$.
• Is connected to a mesh and (possibly) a domain description.

Routines in the FEAT library are usually based on t_blockDiscretisation as this structure resembles the finite element space of the complete equation.

Special case: Also simple equations like the Poisson equation are realised by t_blockDiscretisation although they contain only one finite element space. Such an equation is realised using a 1-block tensor product space $X_h$ = $V_h$ and results in the end in a 1x1 block matrix with 1 block in the corresponding block vectors.

## Uniform and conformal discretisations

Currently, FEAT2 supports uniform and conformal discretisations:

• Uniform discretisations contain only one type of finite element for all cells. The transformation rule is the same for all cells. All cells have the same shape, i.e., "mixed" triangulations (like triangles and quads in the same mesh) are not allowed. Hanging nodes are not allowed.
• Conformal discretisations extend uniform discretisations. It is popssible to mix different type of cells (e.g. triangles and quads) in one mesh. However, hanging nodes are not allowed. All types of finite elements that are used must be "compatible" in terms of degrees of freedom. So it is possible to use a mixed triangle/quad mesh with P1/Q1. However, mixing P1 with Q2 is not possible as the degrees of freedom are not compatible.

Example for a uniform discretisation. All cells are quads and discretised with the same finite element space $Q_1$.

Example for a conformal discretisation. Mixed triangle/quad mesh, discretised with $P_1$ and $Q_1$. The finite element spaces are chosen in such a way that the degrees of freedom match, here, in the vertices.

## Working with finite element spaces

For the creation of finite element spaces, at least a triangulation is needed - this is called rtriangulation in the following. The module spatialdiscretisation.f90 contains a couple of methods to create discretisation structures.

A finite element space is always identified by an "element-ID". All element constants are defined in the module element.f90. The following tables contain an extract of finite element spaces supported by FEAT2.

### 1D elements

Element-ID Type of the element
EL_P0_1D $P_0$, piecewise constant
EL_P1_1D $P_1$, piecewise linear
EL_P2_1D $P_2$, piecewise quadratic

### 2D elements

Element-ID Shape Conforming Parametric Type
EL_P0_2D triangle X X $P_0$, piecewise constant
EL_P1_2D X X $P_1$, piecewise linear
EL_P2_2D X X $P_2$, piecewise quadratic
EL_P3_2D X X $P_3$, piecewise cubic
EL_P1T_2D X Crouzeix-Raviart
EL_Q0_2D quad X X $Q_0$, piecewise constant
EL_Q1_2D X X $Q_1$, piecewise linear
EL_Q2_2D X X $Q_2$, piecewise quadratic
EL_Q3_2D X X $Q_3$, piecewise cubic
EL_QP1_2D X $P_1^\text{disc}$, piecewise linear
EL_E030_2D $\tilde Q_1$, integral mean value based
EL_E031_2D $\tilde Q_1$, midpoint-based
EL_EM30_2D $\tilde Q_1$, integral mean value based
EL_EM31_2D $\tilde Q_1$, midpoint-based

### 3D elements

Element-ID Shape Conforming Parametric Type
EL_P0_3D tetra X X $P_0$, piecewise constant
EL_P1_3D X X $P_1$, piecewise linear
EL_P2_3D X X $P_2$, piecewise quadratic
EL_P3_3D X X $P_3$, piecewise cubic
EL_Q0_3D hex X X $Q_0$, piecewise constant
EL_Q1_3D X X $Q_1$, piecewise linear
EL_Q2_3D X X $Q_2$, piecewise quadratic
EL_Q3_3D X X $Q_3$, piecewise cubic
EL_QP1_2D X $P_1^\text{disc}$, piecewise linear
EL_E030_3D X $\tilde Q_1$, integral mean value based
EL_E031_3D X $\tilde Q_1$, midpoint-based
EL_EM30_3D $\tilde Q_1$, integral mean value based
EL_EM31_3D $\tilde Q_1$, midpoint-based

Using the element-ID, a finite element space can be initialised. The module spatialdiscretisation.f90 contains a set of routines to create an appropriate t_scalarDiscretisation structure:

1. The following command creates a uniform discretisation with $Q_1$ in 2D based on a mesh rtriangulation. THe mesh is expected to contain only quads. The discretisation structure rspatialDiscr identifies the finite element space:

type(t_spatialDiscretisation) :: rspatialDiscr
...
call spdiscr_initDiscr_simple (rspatialDiscr,EL_Q1_2D,rtriangulation)


If the triangulation is releated to an analytical description of the boundary rboundary, this description has to be specified as well:

type(t_spatialDiscretisation) :: rspatialDiscr
...
call spdiscr_initDiscr_simple (rspatialDiscr,EL_Q1_2D,rtriangulation,rboundary)

2. If the mesh contains triangles and quads, a mixed discretisation has to be created. This is done by the following command, where two finite element identifiers are specified. The example creates a mixed 2D finite element space with $P_1$ to be used for triangles and $Q_1$ for quads:

type(t_spatialDiscretisation) :: rspatialDiscr
...
call spdiscr_initDiscr_simple (rspatialDiscr,&
EL_P1_2D,EL_Q1_2D,rtriangulation)


If the triangulation is releated to an analytical description of the boundary rboundary, this description has to be specified as well:

type(t_spatialDiscretisation) :: rspatialDiscr
...
call spdiscr_initDiscr_simple (rspatialDiscr,&
EL_P1_2D,EL_Q1_2D,rtriangulation,rboundary)


To release a finite element space the routine spdiscr_releaseDiscr has to be used:

call spdiscr_releaseDiscr (rspatialDiscr)


## Working with tensor product spaces

Having defined a set of finite element spaces, a tensor product space (identified by a t_blockDiscretisation structure) can be initialised. This procedure requires three steps: Creation of an empty tensor product space, adding all subspaces and finishing the creation.

The following example demonstrates the creation of a tensor product space $X_h = V_h \times W_h$, where $V_h$ and $W_h$ are identified by rspatialDiscrVh and rspatialDiscrWh, respectively. Similar to the creation of a finite element space above, a triangulation and (possibly) a domain definition has to be specified for the creation of the tensor product space:

type(t_spatialDiscretisation) :: rspatialDiscrVh, rspatialDiscrWh
type(t_blockDiscretisation) :: rblockDiscr
...
call spdiscr_initBlockDiscr (rblockDiscr, rtriangulation, rboundary)
call spdiscr_appendBlockComponent (rblockDiscr, rspatialDiscrVh)
call spdiscr_appendBlockComponent (rblockDiscr, rspatialDiscrWh)
call spdiscr_commitBlockDiscr (rblockDiscr)


The tensor product space can be released using spdiscr_releaseBlockDiscr:

call spdiscr_releaseBlockDiscr (rblockDiscr)


Remark: If the tensor product space contains a finite element space multiple times, the creation can be simplified. The following example creates a tensor product space $X_h = V_h \times V_h \times V_h \times W_h$:

type(t_spatialDiscretisation) :: rspatialDiscrVh, rspatialDiscrWh
type(t_blockDiscretisation) :: rblockDiscr
...
call spdiscr_initBlockDiscr (rblockDiscr, rtriangulation, rboundary)
call spdiscr_appendBlockComponent (rblockDiscr, rspatialDiscrVh, 3)
call spdiscr_appendBlockComponent (rblockDiscr, rspatialDiscrWh)
call spdiscr_commitBlockDiscr (rblockDiscr)