**3.4 Finite element spaces**

A **finite element space** Xh can be built on a set of geometric elements and associated finite elements. Its definition depends on the **mesh** Mh of the domain Ω as well as the knowledge of the finite element (K, PK, ΣK) associated with each domain K ∈ Mh [6]

Given a function u defined in Ω, regular enough, its interpolant uh ∈ Xh is uniquely defined such as [6]:


A **finite element space** Xh can be built on a set of geometric elements and associated finite elements. Its definition depends on the **mesh** Mh of the domain Ω as well as the knowledge of the finite element (K, PK, ΣK) associated with each domain K∈ Mh

Given a function u defined in Ω, regular enough, its interpolant uh ∈ Xh is uniquely defined such as [6]:


Some continuous conditions have to be ensured across the interfaces between geometric elements, which is the property of conformity.

A **mesh** Mh of the studied domain Ω is defined as a collection of geometric elements which have in common either a facet, or an edge, or a node, or nothing (**Figure 5**). The elements cannot overlap each other.

The finite element space Xh has a finite dimension, denoted Dh. It can be characterized by a set of degrees of freedom Σ<sup>h</sup> linked up to the sets ΣK,∀K∈ Mh, that is

$$\Sigma\_{\mathbf{h}} = \left\{ \Phi\_{\mathbf{h}, \mathbf{j}}, \quad \mathbf{1} \le \mathbf{j} \le \mathbf{D}\_{\mathbf{h}} \right\}.$$

**Figure 5.** *Mesh of a part of a two-dimensional domain Ω.*

It is also possible to define the base functions ph,j, 1 ≤ i ≤ Dh, of the space Xh from the base functions of the spaces PK, ∀ K∈ Mh. Those have to verify the relations

$$\phi\_{\mathbf{j}}\left(\mathbf{p}\_{\mathbf{j}}\right) = \delta\_{\vec{\mathbf{j}}}, \mathbf{1} \le \mathbf{i}, \mathbf{j} \le \mathbf{n}\_{\mathbb{K}} \tag{32}$$

similar to relations (32). They are actually piecewise defined and their supports are as "small" as possible, that is, are constituted by a limited number of geometric elements.

Then, with any function u regular enough so that the degrees of freedom ϕh,j(u), 1 ≤ j ≤ Dh, are well defined, it can be associated a function uh, called **X**h**-interpolant**, defined by

$$\mathbf{u}\_{\mathbf{h}} = \sum\_{\mathbf{j}=1}^{\text{D}\_{\mathbf{h}}} \boldsymbol{\Phi}\_{\mathbf{h},\mathbf{j}}(\mathbf{u}) \mathbf{p}\_{\mathbf{h},\mathbf{j}} \tag{33}$$

#### **4. Construction of a sequence of finite element spaces**

#### **4.1 Geometric elements**

A mesh of a domain is considered which is built with a collection of geometric elements which can be tetrahedra (4 nodes), hexahedra (8 nodes) and prisms (6 nodes) (**Figure 6**) [4, 6, 9].

These elements are called volumes and their vertices represent nodes. The sets of nodes, edges, facets and volumes of this mesh are denoted by N, E, F and V, respectively. Their sizes are #N, #E, #F and #V.

The i-th node of the mesh is denoted by ni or {i}. The edges and facets can be defined with ordered sets of nodes. An edge is denoted by eij or {i, j}, a triangular facet by fijk or {i, j, k}, and a quadrangular facet by fijkl or {i, j, k, l}. These geometric entities are shown in **Figure 7**.

**Figure 6.** *Collection of different geometric elements [6].*

*The Finite Element Method Applied to the Magnetostatic and Magnetodynamic Problems DOI: http://dx.doi.org/10.5772/intechopen.93696*

**Figure 7.** *Geometric entities: Node, edge and facets (i, j, k, l N) [6].*
