**2.2 Green formulas**

The following notations are used for integration of products of scalar or vector fields over a volume Ω or on a surface Γ, where L<sup>2</sup> and **L**<sup>2</sup> are the spaces of squaresummable scalar and vector functions [2, 3]:

$$(\mathbf{u}, \mathbf{v}) = \int\_{\Omega} \mathbf{u}(\mathbf{x}) \mathbf{v}(\mathbf{x}) d\mathbf{x}, \quad \mathbf{u}, \mathbf{v} \in \mathbf{L}^2(\Omega)$$

$$(\boldsymbol{\mathfrak{u}}, \boldsymbol{\mathfrak{v}}) = \int\_{\Omega} \boldsymbol{\mathfrak{u}}(\mathbf{x}) \cdot \boldsymbol{\mathfrak{v}}(\mathbf{x}) d\mathbf{x}, \quad \boldsymbol{\mathfrak{u}}, \boldsymbol{\mathfrak{v}} \in L^2(\Omega),$$

$$\langle \mathbf{u}, \mathbf{v} \rangle\_{\Gamma} = \int\_{\Gamma} \mathbf{u}(\mathbf{x}) \cdot \mathbf{v}(\mathbf{x}) d\mathbf{s}, \quad \langle \mathbf{u}, \mathbf{v} \rangle\_{\Gamma} = \int\_{\Gamma} \mathbf{u}(\mathbf{x}) \cdot \mathbf{v}(\mathbf{x}) d\mathbf{s},$$

A first relation of vectorial analysis

$$
\mathbf{u} \bullet \mathbf{grad} \mathbf{v} + \mathbf{v} \bullet \mathrm{div} \, \mathbf{u} = \mathrm{div} \, (\mathbf{v} \, \mathbf{u}),
$$

integrated in the domain Ω, after applying the divergence theorem, gives the Green formula said of kind grad-div in Ω, that is

$$(\mathfrak{u}, \operatorname{grad} v) + (\operatorname{div} \mathfrak{u}, v) = \_{\Gamma}, \forall \mathfrak{u} \in H^1(\Omega), \forall v \in H^1(\Omega) \tag{5}$$

where *H*<sup>1</sup> ð Þ <sup>Ω</sup> and *<sup>v</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>1</sup> ð Þ Ω are function spaces built for scalar and vector fields, respectively.

Another relation of vectorial analysis

$$
\mathbf{u}.\mathbf{curl}\,\boldsymbol{\nu}-\mathbf{curl}\,\boldsymbol{\omega}\cdot\boldsymbol{\nu} = \mathrm{div}\,(\boldsymbol{\nu}\times\boldsymbol{\mathfrak{u}})\tag{6}
$$

integrated in the domain Ω, after applying the divergence theorem, gives the Green formula said of kind curl-curl in Ω, that is

$$(\mathfrak{u}, \operatorname{curl} \mathfrak{v}) - (\operatorname{curl} \mathfrak{u}, \mathfrak{v}) = <\mathfrak{u} \times \mathfrak{v}, \mathfrak{v} >\_{\Gamma} . \quad \forall \mathfrak{u}, \mathfrak{v} \in H^1(\Omega) \tag{7}$$

Note that the surface integral term appearing in this last formula can take the following similar forms:

$$<\langle \mathfrak{u} \times \mathfrak{n}, \mathfrak{v} >\_{\Gamma} = \langle \mathfrak{v} \times \mathfrak{u}, \mathfrak{n} \rangle\_{\Gamma} = -\langle \mathfrak{v} \times \mathfrak{n}, \mathfrak{u} >\_{\Gamma} \rangle$$

It is possible to define a **generalized Green formula** by

$$\mathbf{u}(\mathbf{L}\mathbf{u}, \mathbf{v}) - (\mathbf{u}, \mathbf{L}^\* \mathbf{v}) = \int\_{\varGamma} \mathbf{Q}(\mathbf{u}, \mathbf{v}) \mathbf{ds}, \forall \mathbf{u} \in \text{dom}(\mathbf{L}) \text{ and} \forall \mathbf{v} \in \text{dom}\,(\mathbf{L}^\*), \tag{8}$$

where L and L\* are differential operators of order n which act respectively on functions u and v defined in Ω, with Ω = Ω ∪ Γ; Q is a bi-linear function of u and v. The operator L\* is called the **dual operator** of L. It can easily be seen that formulas (6) and (7) are particular cases of (8).

#### **2.3 Weak formulations**

Consider a partial differential problem of the form [4].

$$\mathbf{L}\,\mathbf{u} = \mathbf{f}\,\text{in}\,\Omega,\tag{9}$$

$$\mathbf{B} \cdot \mathbf{u} = \mathbf{g} \text{ on } \Gamma = \partial \Omega,\tag{10}$$

where L is a differential operator of order n, B is an operator which defines a boundary condition, f and g are functions respectively defined in Ω and on its boundary Γ, and u is an unknown function from a function space U and defined in Ω, that is, u ∈ U(Ω). Note that f can eventually depend on u.

Problems (9 and 10) constitute what is called a **classical formulation**, or strong formulation. A function u ∈ U(Ω) which verifies this problem is called a **classical solution**, or strong solution. Particularly, as L is of order n, the function u has to be n–1 times continuously differentiable, that is, u ∈ Cn–1(Ω).

A **weak formulation** of problem (9) is defined as having the generalized form.

$$(\mathbf{u}, \mathbf{L}^\* \mathbf{v}) - (\mathbf{f}, \mathbf{v}) + \int\_{\Gamma} \mathbf{Q}\_{\mathbf{g}}(\mathbf{v}) \mathbf{ds} = \mathbf{0}, \quad \forall \mathbf{v} \in \mathbf{V}(\Omega) \tag{11}$$

where L\* is the dual operator of L, defined by the generalized Green formula (8), Qg is a linear form in v which depend on g, and the space V (Ω) is a space of **test functions** which has to be defined according to the operator L\* and particularly according to the boundary condition (9 and 10). A function u which satisfies this equation for any test function v ∈ V (Ω) is called a **weak solution**.

The generalized Green formula (8) can be applied to formulation (11) in order to get L instead of L\*, which usually consists of performing an integration by parts. It is then possible to find again, thanks to a judicious choice of test functions, the equations and relations of the classical formulation of the problem, that is, Eq. (9) and boundary condition (11).

It is often easy to check that a classical solution is also a weak solution. Nevertheless, it is not always straightforward that a weak solution is also a classical solution because it has to be regular enough in order to be defined at the classic sense.

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

One mathematical advantage of weak formulations is that they usually enable to prove the existence of a solution easier than classical formulations do. The solution has then to be proved to be regular enough to be also a classical solution. Another advantage of weak formulations is that they are well adapted to a discretization using finite elements and then to a numerical solution, which is not the case with classical formulations.

In some cases, it is possible to define a minimization problem similar to the weak formulation (11).

#### **2.4 A weak formulation for the magnetodynamic problem**

In order to illustrate the notion of weak formulation, consider the magnetodynamic problem, limited to the domain Ω, with boundary ∂Ω = Γ = Γ<sup>h</sup> ∪ Γ<sup>e</sup> (**Figure 2**), whose equations and material relations are written in Euclidean space <sup>3</sup> [5, 6].

$$\text{curl } \mathbf{h} = \mathbf{j}\_s \tag{12}$$

$$\text{curl } \mathbf{e} = -j\boldsymbol{\omega} \cdot \mathbf{b} \tag{13}$$

$$\operatorname{div} \mathbf{b} = \mathbf{0} \tag{14}$$

with behavior relations of materials.

$$\mathbf{b} = \mu \mathbf{h} \tag{15}$$

$$
\dot{j} = \sigma \mathbf{e},
\tag{16}
$$

where *j* is called the imaginary unit, *b* is the magnetic induction (T), *e* is the electric field (V/m), *j <sup>s</sup>* is the current density (A/m2), *h* is the magnetic field (A/m), *μ* is the relative permeability and *σ* is the electric conductivity (S/m). From the Eq. (13), the field *b* can be obtained from a magnetic vector potential *a<sup>i</sup>* via the term:

$$
\mathfrak{b} = \text{curl } \mathfrak{a}.\tag{17}
$$

Combining (15 and 16) into (14), one has curl (*<sup>e</sup>* <sup>þ</sup> *<sup>∂</sup>*t*a*Þ ¼ 0, that leads to the presentation of an electric scalar potential *<sup>ν</sup>* through *<sup>e</sup>* ¼ �*∂ta* � grad *<sup>υ</sup>*.

By starting from the Ampere's law (12), the weak form of magnetic vector potential is written as [4, 6].

**Figure 2.** *Studied domain Ω and its boundary.*

$$\left(\mu^{-1}\mathbf{b}, \mathbf{curl}\,\mathbf{a}'\right)\_{\Omega} - \left(\sigma\mathbf{e}, \mathbf{a}'\right)\_{\Omega\_{\mathbf{e}}} + \left(\mathbf{n} \times \mathbf{b}, \mathbf{a}'\right)\_{\Gamma} = \left(\mathbf{j}\_{\mathbf{s}}, \mathbf{a}'\right)\_{\Omega\_{\mathbf{e}}} \forall \mathbf{a}' \in \mathsf{H}\_{\mathbf{e}}^{0}(\mathbf{curl}, \Omega) \tag{18}$$

Combining the magnetic vector potential *a* and the electrical field *e*, one has

$$\begin{aligned} & \left( \mu^{-1} \text{curl } \mathfrak{a}, \text{curl } \mathfrak{a}' \right)\_{\mathfrak{Q}} + \left( \sigma \partial\_{t} \mathfrak{a}, \mathfrak{a}' \right)\_{\mathfrak{Q}\_{t}} + \left( \sigma \text{grad } \mathfrak{v}, \text{curl } \mathfrak{a}' \right)\_{\mathfrak{Q}\_{t}} + \left< \mathfrak{n} \times \mathfrak{h}, \mathfrak{a}' \right>\_{\Gamma\_{h}} \\ &= \left( \mathfrak{j}\_{i}, \mathfrak{a}' \right)\_{\mathfrak{Q}\_{t}}, \forall \mathfrak{a}'\_{i} \in F^{0}\_{\mathfrak{c}}(\text{curl}, \mathfrak{Q}\_{i}), \end{aligned} \tag{19}$$

where *H*<sup>0</sup> *<sup>e</sup>* ð Þ curl, Ω is a function space defined on Ω containing the basis functions for *a* as well as for the test function *a*<sup>0</sup> (at the discrete level, this space is defined by edge FEs; notations (�, �) and < �, � > are respectively a volume integral in and a surface integral of the product of their vector field arguments.

#### **2.5 A weak formulation for the magnetostatic problem**

The magnetostatic problem is considered as a simplification of the magnetodynamic formulation where all time dependent phenomena are neglected. In a same way, by starting from the Ampere's law (12), this initial form of the problem is its **classical formulation**.

Consider the Green formula of type grad-div in Ω (5) applied to the field *b* and to a scalar field ϕ' to be defined, that is [6–8]

$$(\mathbf{b}, \operatorname{grad} \phi') + (\operatorname{div} \mathbf{b}, \phi') = \langle \mathbf{n} \mid \mathbf{b}, \phi' >\_{\Gamma}, \forall \phi' \in \Phi \; (\Omega). \tag{20}$$

If the space Φ(Ω) is defined such as

$$\Phi(\Omega) = \{ \phi \in \mathbf{H1}(\Omega); \phi | \Gamma\_{\mathrm{h}} = \mathbf{0} \}, \tag{21}$$

then the last term of Eq. (20) is reduced to < *n*.*b*, ϕ' > Γ<sup>e</sup> and is equal to zero if condition (15) is taken into account. Moreover, the second term of this equation is equal to zero because of Eq. (16). Eq. (20) can then be reduced to.

$$(\mathfrak{b}, \mathfrak{grad}\,\phi') = \mathfrak{0}, \forall \phi' \in \Phi(\mathfrak{Q}).\tag{22}$$

This last form is called a **weak formulation** of the problem. It has been established starting from a Green formula but it can be considered now as an a priori posed form whose enclosed information can be deduced.

In fact, weak formulation (22) contains both **Eq.** (12) and **boundary condition** (15). Indeed, by applying the Green formula of type grad-div to it, we get.

$$(\operatorname{div}\mathbf{b}, \boldsymbol{\Phi}') = \langle \mathfrak{n} \bullet \mathfrak{b}, \boldsymbol{\Phi}' \rangle\_{\Gamma}, \forall \boldsymbol{\Phi}' \in \boldsymbol{\Phi}(\boldsymbol{\Omega}).\tag{23}$$

This equation is verified for any test function ϕ' ∈ Φ(Ω) and thus, particularly, for any function ϕ' whose value is equal to zero on Γ, that is, ϕ' ∈ Φ<sup>0</sup> (Ω) because Φ0(Ω) ⊂ Φ(Ω). Therefore, it comes that (div *b*, ϕ') = 0 for any function ϕ' of this kind and, consequently, that div **b** = 0 in Ω, that is, Eq. (12) is satisfied. Then, Eq. (23) is reduced to ⟨*n* ∙ *b*, ϕ<sup>0</sup> ><sup>Γ</sup> = 0, and by considering now all the functions ϕ' ∈ Φ(Ω) without any restriction, that is, which can vary freely on Γ*e*, it comes that *n* ∙ *b*j Γ*e* = 0, that is, that condition (13) is satisfied.

It is possible to obtain more information from the weak formulation, particularly as far as the **transmission conditions** on surfaces inside Ω are concerned. Consider for that two subdomains Ω<sup>1</sup> and Ω<sup>2</sup> of Ω separated by an interface Σ (**Figure 3**) [7].

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

**Figure 3.** *Interface Σ between two subdomains Ω<sup>1</sup> and Ω2.*

Let us apply the Green formula of type grad-div (5) to the fields *b* and ϕ' successively in both subdomains Ω<sup>1</sup> and Ω2. By taking into account that div *b* = 0 in Ω, and thus particularly in Ω<sup>1</sup> and Ω2, then by summing the obtained relations, we get the relation [6, 7].

$$(\mathfrak{b}, \operatorname{grad} \phi')\Omega\_1 \approx \Omega\_2 = <\mathfrak{n}.\\(\mathfrak{b}\_1 \mathfrak{b}\_2), \phi' > \Sigma + <\mathfrak{n}. \mathfrak{b}, \phi' > (\mathfrak{Q}\_1 \approx \Omega\_2), \forall \phi' \in \Phi(\mathfrak{Q}), \tag{24}$$

where *b<sup>1</sup>* and *b<sup>2</sup>* represent the field *b* on both sides of Σ in the respective domains Ω1 and Ω2. Considering the test functions ϕ' whose support is Ω1∪Ω<sup>2</sup> and which are equal to zero on \_(Ω1∪Ω2), it remains from (24) the well known transmission condition *n.(b1–b2)*∣Σ = 0. Note that the first term of (24) vanishes thanks to Eq. (22) indeed, the domain of integration Ω<sup>1</sup> ∪ Ω<sup>2</sup> can be extended to Ω thanks to the chosen test functions.

The way to establish a weak formulation of Eq. (13) has been described and the richness of the information enclosed in such a formulation has been shown up. Using a similar procedure, a weak formulation associated with Eq. (12) can be established, but we will proceed differently in order to keep some classical equations.

If the field *h* is decomposed into a given source component *hs*, such as curl *h<sup>s</sup>* = *j*, and a reaction component *hr*, then curl *h<sup>r</sup>* = 0 and *h<sup>r</sup>* is therefore of the form *h<sup>r</sup>* = � grad ϕ (if Ω is simply connected). This consists of satisfying Eq. (15) classically. Taking into account the behavior law (15), we can write *b* = μ (*hs* � grad ϕ) and put this last expression in (24) to obtain.

$$(\mu \left(\hbar, -\operatorname{grad}\Phi\right), \operatorname{grad}\Phi') = 0, \forall \phi' \in \Phi(\Omega). \tag{25}$$

This formulation contains the whole problem (12 and 13). The potential ϕ is the unknown and all the other fields can be deduced from ϕ thanks to the equations which have been kept on a classical form. It appears that the potential ϕ belongs to the same space of the test functions or at least to a space Φ<sup>r</sup> (Ω) which is parallel to it, that is, where the boundary condition relative to ϕ on Γh is not necessarily homogeneous, that is, ϕ∣Γ<sup>h</sup> = constant. Note that this boundary condition on Γ<sup>h</sup> is implicitly taken into account in the space Φ(Ω).

Weak formulation (25) can be considered as **a system of an infinite number of equations with an infinite number of unknowns**. It will be seen in the following how such a problem can be approximated to lead to a numerical solution. This approximation will constitute the phase of discretization.

A similar minimization problem

It is possible to define a **minimization problem** associated with (25). For that, let us define the functional [2, 3].

$$\mathbf{W}(\boldsymbol{\Phi}) = (\mu(\mathbf{hs} - \mathbf{grad}\,\boldsymbol{\Phi}), \mathbf{h}\_t - \mathbf{grad}\,\boldsymbol{\Phi}), \tag{26}$$

and let us pose the following minimization problem:

$$\text{find } \phi \in \Phi\_{\mathbf{r}}(\mathfrak{Q}) \text{ such as } \mathbf{W}(\phi) \le \mathbf{W}(\phi') , \forall \phi' \in \Phi \mathbf{r}(\mathfrak{Q}). \tag{27}$$

The physical materials are considered having linear magnetic behavior, but the following can be generalized easily for nonlinear materials.

By stationarizing functional (25) in relation to ϕ, it can be verified that (25) is obtained. It can also be verified that the solution ϕ of (25) minimizes this functional. Indeed, let us suppose that ϕ ∈ Φr(Ω) is solution of (25) and let us consider any function ψ ∈ Φr(Ω); then let us pose η = ψ � ϕ, which implies η ∈ Φ(Ω); we have

$$\mathbf{W}(\boldsymbol{\Psi}) = \mathbf{W}(\boldsymbol{\Phi} + \boldsymbol{\eta}) = (\boldsymbol{\mu} \ (\mathbf{h}\_{\boldsymbol{\varepsilon}} - \mathbf{grad} \ (\boldsymbol{\phi} + \boldsymbol{\eta})), \boldsymbol{h}\_{\boldsymbol{\varepsilon}} - \mathbf{grad} \ (\boldsymbol{\phi} + \boldsymbol{\eta})).$$

and thus.

$$\mathbf{W}(\boldsymbol{\Psi}) = \mathbf{W}(\boldsymbol{\Phi}) + (\boldsymbol{\mu}\,\mathbf{grad}\,\boldsymbol{\eta}, \mathbf{grad}\,\boldsymbol{\eta}) + (\boldsymbol{\mu}(\mathbf{h}\_{\boldsymbol{\nu}} - \mathbf{grad}\,\boldsymbol{\Phi}), -\mathbf{grad}\,\boldsymbol{\eta}).$$

As the second term of this sum is positive or equal to zero and the third term is equal to zero, because of (25), it comes that W(ψ) and W(ϕ).

Formulations (25) and (27) are then similar. Note that W(ϕ) is the magnetic coenergy and that the problem actually consists of minimizing this coenergy.

If the continuous function spaces are replaced by **discrete spaces**, and if the considered test functions are limited to these spaces, **then the information inside a weak formulation will only be satisfied approximately**, or weakly.

The basis of the discretization of weak formulations can be illustrated for the above magnetostatic problem, whose weak formulation is (25), that is

$$(\mu(\mathfrak{h}\_{\mathfrak{t}} - \operatorname{grad} \, \mathfrak{\phi}), \operatorname{grad} \, \mathfrak{\phi}') = \mathbf{0}, \forall \mathfrak{\phi}' \in \Phi(\mathfrak{\Omega}), \tag{28}$$

with ϕ ∈ Φ(Ω). The space Φ(Ω) has to be replaced by a discrete space Φh(Ω) which is a subset of it, that is, Φh(Ω) ⊂ Φ(Ω). This space has a finite dimension, denoted N, and can then be defined by N linearly independent base functions. The principle is then to look for the function ϕ in Φ<sup>h</sup> (Ω), which consists of determining N unknown parameters. This function will be only an approximation of the exact solution ϕ ∈ Φ (Ω). The more the functions of Φ<sup>h</sup> (Ω) approximate well those of Φ (Ω), the higher the quality of the approximation is. Each test function ϕ' will lead to an equation of the form (28) and, as the number of equations and unknowns has to be the same, N linearly independent test functions have to be chosen. This choice can be made on the base functions of Φ<sup>h</sup> (Ω) and the method is called the **Galerkine method**. Such base functions are defined thanks to finite elements.

#### **3. Finite elements**

#### **3.1 Definition of a finite element**

A **finite element** is defined by the **three element set (K, P**K**, Σ**K**)** where [2, 6, 7]:


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

• ΣK is a set of nK **degrees of freedom** represented by nK linear functionals ϕi, 1 ≤ i ≤ nK, defined in space PK;

moreover, any function u ∈ PK must be defined uniquely by the degrees of freedom of ΣK, which defines the unisolvance of the finite element (K, PK, ΣK).

The role of a finite element is to interpolate a field in a function space of finite dimension. Several finite elements can be defined on the same geometric element and then, under certain conditions, can form mixed finite elements. **Figure 4** shows the various spaces which occur in the definition of a finite element; the definition of the subspace of points κ ⊂ K is actually associated with the definition of the functionals.

For the most commonly used finite elements, the degrees of freedom are associated with nodes of K and the functionals ϕi are reduced to functions of the coordinates in K; these elements are called **nodal finite elements**. Nevertheless, the above definition is more general thanks to the freedom let in the choice of the functionals. It will be shown that these can be, in addition to nodal values, integrals along segments, on surfaces or in volumes; the subspace of points κ ⊂ K (**Figure 4**) is then respectively a point, a segment, a surface or a volume.

#### **3.2 Unisolvant finite element**

The finite element (K, PK, ΣK) is **unisolvant** if [6].

$$\forall \mathbf{p} \in \mathbf{P}\_{\mathbf{K}}, \phi \mathbf{i}(\mathbf{p}) = \mathbf{0}; \forall \phi \mathbf{i} \in \Sigma \mathbf{K} \Rightarrow \mathbf{p} \equiv \mathbf{0}.$$

In this case, for any function u regular enough, one can define a **unique interpolation** uK, called **PK-interpolant**, such as.

$$
\phi \mathbf{i} (\mathbf{u} - \mathfrak{u}\_{\mathsf{K}}) = \mathbf{0}, \forall \phi \mathbf{i} \in \Sigma\_{\mathsf{K}}; \mathfrak{u}\_{\mathsf{K}} \in \mathsf{P}\_{\mathsf{K}}.\tag{29}
$$

The set Σ<sup>K</sup> is said PK - unisolvant.

Proof:

Each function p ∈ PK can be written as a linear combination of functions of a base of PK, denoted {pi, 1 ≤ i ≤ nK}, that is

$$\mathbf{p} = \sum\_{i=1}^{\mathfrak{n}\_{\mathbb{K}}} \mathbf{a}\_{i} \mathbf{p}\_{i},$$

where the pi, 1 ≤ i ≤ nK, are called **base functions**.

**Figure 4.** *Spaces associated with a finite element (K, PK, ΣK) [6].*

As the functionals ϕj, 1 ≤ j ≤ nK, are linear, we have

$$\phi\_{\mathbf{j}}(\mathbf{p}) = \sum\_{i=1}^{n\_{\mathbb{K}}} \mathbf{a}\_{i} \phi\_{\mathbf{j}}(\mathbf{p}\_{i}), \mathbf{1} \le \mathbf{j} \le \mathbf{n}\_{\mathbb{K}}.$$

And, as ϕj(p) = 0, 1 ≤ j ≤ nK, leads to p � 0, the determinant of the matrix Φ (Φji = ϕj(pj), 1 ≤ i, j ≤ nK) is not equal to zero; indeed the solution of the corresponding system must be identically equal to zero (i.e., ai = 0, 1 ≤ i ≤ nK). Consequently, the system

$$\mathfrak{d}\_{\mathfrak{j}}(\mathbf{u}) = \mathfrak{d}\_{\mathfrak{j}}(\mathbf{u}\_{\mathbb{K}}) \Leftrightarrow \mathfrak{d}\_{\mathfrak{j}}(\mathbf{u}) = \sum\_{i=1}^{n\_{\mathbb{K}}} \mathfrak{a}\_{i} \mathfrak{d}\_{\mathfrak{j}}(\mathbf{p}\_{i}), \mathbf{1} \leq \mathfrak{j} \leq \mathbf{n}\_{\mathbb{K}}.$$

has a unique solution (aj , 1 ≤ i ≤ nK).

#### **3.3 Degrees of freedom**

The interpolation of a function u, in the space PK and in K, is given by the expression [4]

$$\mathbf{u}\_{\mathbf{K}} = \sum\_{i=1}^{n\_{\mathbf{K}}} \mathbf{a}\_{i} \mathbf{p}\_{i}, \mathbf{u}\_{\mathbf{K}} \in \mathbf{P}\_{\mathbf{K}},$$

where the nK coefficients aj associated with the base functions pj ∈ PK can be determined thanks to relations (26), that is, thanks to the solution of the linear system.

$$\phi\_{\mathfrak{j}}(\mathbf{u}) = \sum\_{i=1}^{n\_{\mathbb{K}}} \mathbf{a}\_{i} \phi\_{\mathfrak{j}}(\mathbf{p}\_{i}), \mathbf{1} \le \mathbf{j} \le \mathbf{n}\_{\mathbb{K}}.$$

provided that the function u is sufficiently regular for the ϕj(u), 1 ≤ j ≤ nK, to exist.

This solution is simplified to the maximum if we define the functionals so that

$$\phi\_{\mathbf{j}}\left(\mathbf{p}\_{\mathbf{j}}\right) = \delta\_{\mathbf{j}}, \mathbf{1} \le \mathbf{i}, \mathbf{j} \le \mathbf{n}\_{\mathbf{K}} \tag{30}$$

where δi,j is the Kronecker symbol, that is

$$\mathfrak{d}\_{\vec{\mathbf{u}}} = \begin{cases} \mathbf{1} \, \mathbf{s} \, \vec{\mathbf{u}} = j \\ \mathbf{0} \, \mathbf{s} \, \vec{\mathbf{u}} \neq j \end{cases}$$

The matrix of the system is then the unit matrix and the solution is

$$\mathbf{a\_j} = \boldsymbol{\Phi\_j}(\mathbf{u}), \mathbf{1} \le \mathbf{j} \le \mathbf{n\_K}$$

In this case, the interpolation uK ∈ PK is expressed by

$$\mathbf{u}\_{\mathbf{K}} = \sum\_{\mathbf{j=1}}^{\mathbf{n}\_{\mathbf{K}}} \boldsymbol{\phi}\_{\mathbf{j}}(\mathbf{u}) \mathbf{p}\_{\mathbf{j}},\tag{31}$$

where the coefficients ϕj(u) = ϕj(uK), 1 ≤ j ≤ nK, are called **degrees of freedom**.

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