**2.5 The FE method using Lagrange <sup>2</sup> elements**

Let *a* ¼ *x*<sup>0</sup> <*x*<sup>1</sup> < ⋯ <*xN* ¼ *b* be a regular partition of the interval ½ � *a*, *b* . Suppose that the length of *Ii* <sup>¼</sup> *xi*�1, *xi* ½ � is *hi* <sup>¼</sup> *xi* � *xi*�1. Let *<sup>P</sup><sup>k</sup>* <sup>¼</sup> *p x*ð Þ¼ <sup>P</sup>*<sup>k</sup> <sup>j</sup>*¼<sup>0</sup>*cjx <sup>j</sup>* ,*cj* ∈ n o denotes the vector space of polynomials in one variable and of degree less than or equal to *k*. The FE method for Lagrange *P*<sup>2</sup> elements involves the discrete space:

$$V\_h^2 = \{ \boldsymbol{\nu}(\boldsymbol{\kappa}) \in \mathbb{C}^0[\boldsymbol{a}, b], \quad \boldsymbol{\nu}|\_{I\_i} \in P^2(I\_i), \quad i = 1, \ldots, N \},$$

and its subspace *V*<sup>2</sup> 0,*<sup>h</sup>* <sup>¼</sup> *<sup>v</sup>*∈*V*<sup>2</sup> *<sup>h</sup>*<sup>j</sup> *v a*ð Þ¼ *v b*ð Þ¼ <sup>0</sup> � �*:* These spaces are composed of continuous, piecewise parabolic functions (polynomials of degree less than or equal to 2). The *P*<sup>2</sup> FE method consists in applying the internal variational approximation approach to these spaces.

**Lemma 2.2** *The space V*<sup>2</sup> *<sup>h</sup> is a subspace of H*<sup>1</sup> ½ � *a*, *b of dimension* 2*N* þ 1*. Every function vh* ∈*V*<sup>2</sup> *<sup>h</sup> is uniquely defined by its values at the mesh vertices x <sup>j</sup>*, *j* ¼ 0, 1, … , *N and at the midpoints x <sup>j</sup>*þ<sup>1</sup> <sup>2</sup> <sup>¼</sup> *<sup>x</sup> <sup>j</sup>*þ*<sup>x</sup> <sup>j</sup>*þ<sup>1</sup> <sup>2</sup> <sup>¼</sup> *<sup>x</sup> <sup>j</sup>* <sup>þ</sup> *<sup>h</sup> <sup>j</sup>*þ<sup>1</sup> <sup>2</sup> , *j* ¼ 0, 1, … , *N* � 1*, where h <sup>j</sup>*þ<sup>1</sup> ¼ *x <sup>j</sup>*þ<sup>1</sup> � *x j:*

$$v\_h(\mathbf{x}) = \sum\_{j=0}^{N} v\_h(\mathbf{x}\_j) \phi\_j(\mathbf{x}) + \sum\_{j=0}^{N-1} v\_h\left(\mathbf{x}\_{j+\frac{1}{2}}\right) \phi\_{j+\frac{1}{2}}(\mathbf{x}), \quad \forall \ \mathbf{x} \in [a, b], \mathbf{y}$$

where *ϕ<sup>j</sup>* n o*<sup>N</sup> <sup>j</sup>*¼<sup>0</sup> is the basis of the shape functions *<sup>ϕ</sup><sup>j</sup>* defined as:

$$\begin{aligned} \phi\_j(\mathbf{x}) &= \phi\left(\frac{\mathbf{x} - \mathbf{x}\_j}{h\_{j+1}}\right), \quad j = \mathbf{0}, \mathbf{1}, \dots, N, \quad \phi\_{j + \frac{1}{2}}(\mathbf{x}) = \nu\left(\frac{\mathbf{x} - \mathbf{x}\_{j + \frac{1}{2}}}{h\_{j+1}}\right), \\\ j &= \mathbf{0}, \mathbf{1}, \dots, N - \mathbf{1}, \end{aligned}$$

with

$$\phi(\xi) = \begin{cases} (1+\xi)(1+2\xi), & \xi \in [-1,0], \\ (1-\xi)(1-2\xi), & \xi \in [0,1], \\ 0, & |\xi| > 1, \end{cases} \quad \psi(\xi) = \begin{cases} 1-4\xi^2, & |\xi| \le \frac{1}{2}, \\ 0, & |\xi| > \frac{1}{2}, \end{cases} \tag{28}$$

**Figure 3** shows the global shape functions for the space *V*<sup>2</sup> *<sup>h</sup>* and the three quadratic Lagrange *<sup>P</sup>*<sup>2</sup> shape functions on the reference interval ½ � �1, 1 .

*A Brief Summary of the Finite Element Method for Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.95423*

#### **Figure 3.**

*(left) global shape functions for the space V*<sup>2</sup> *h. (right) the three quadratic Lagrange P*<sup>2</sup> *shape functions on the reference interval* ½ � �1, 1 *.*

**Remark 2.5** *Notice that we have:*

$$\phi\_j(\mathbf{x}\_j) = \delta\_{\vec{\eta}}, \quad \phi\_j\left(\mathbf{x}\_{j+\frac{1}{2}}\right) = \mathbf{0}, \quad \phi\_{j+\frac{1}{2}}(\mathbf{x}\_j) = \mathbf{0}, \quad \phi\_{j+\frac{1}{2}}\left(\mathbf{x}\_{j+\frac{1}{2}}\right) = \delta\_{\vec{\eta}}.$$

**Corollary 2.1** *The space V*<sup>2</sup> 0,*<sup>h</sup> is a subspace of H*<sup>1</sup> <sup>0</sup>½ � *a*, *b of dimension* 2*N* � 1 *and every function vh* ∈*V*<sup>2</sup> 0,*<sup>h</sup> is uniquely defined by its values at the mesh vertices x <sup>j</sup>*, *j* ¼ 1, 2, … , *<sup>N</sup>* � <sup>1</sup> *and at the midpoints x <sup>j</sup>*þ<sup>1</sup> 2 , *j* ¼ 0, 1, … , *N* � 1*:*

$$v\_h(\mathbf{x}) = \sum\_{j=1}^{N-1} v\_h(\mathbf{x}\_j) \phi\_j(\mathbf{x}) + \sum\_{j=0}^{N-1} v\_h(\mathbf{x}\_{j+\frac{1}{2}}) \phi\_{j+\frac{1}{2}}(\mathbf{x}), \quad \forall \ x \in [a, b], \mathbf{y}$$

where *ϕ<sup>j</sup>* n o*<sup>N</sup> <sup>j</sup>*¼<sup>0</sup> is the basis of the shape functions *<sup>ϕ</sup><sup>j</sup>* defined as:

$$\phi\_j(\mathbf{x}) = \phi\left(\frac{\mathbf{x} - \mathbf{x}\_j}{h\_{j+1}}\right), \quad j = 0, 1, \dots, N, \quad \phi\_{j + \frac{1}{2}}(\mathbf{x}) = \varphi\left(\frac{\mathbf{x} - \mathbf{x}\_{j + \frac{1}{2}}}{h\_{j+1}}\right), \quad j = 0, 1, \dots, N - 1, \varphi$$

with *ϕ ξ*ð Þ and *ψ ξ*ð Þ are defined by (28).

#### *2.5.1 Homogeneous boundary conditions*

The variational formulation of the internal approximation of the Dirichlet BVP (3) consists now in finding *uh* ∈*V*<sup>2</sup> 0,*<sup>h</sup>*, such that:

$$\int\_{a}^{b} u'\_h v' d\mathfrak{x} + \int\_{a}^{b} qu\_h v d\mathfrak{x} = \int\_{a}^{b} fv d\mathfrak{x}, \quad \forall \ v \in V\_{h,0}^2.$$

Here, it is convenient to introduce the notation *x <sup>j</sup>* 2 , *j* ¼ 1, … , 2*N* � 1 for the mesh points and *ϕ <sup>j</sup>* 2 , *<sup>j</sup>* <sup>¼</sup> 1, … , 2*<sup>N</sup>* � 1 for the basis of *<sup>V</sup>*<sup>2</sup> 0,*<sup>h</sup>*. Using these notations, we have:

$$
\mu\_h = \sum\_{j=1}^{2N-1} c\_{\frac{j}{2}} \phi\_{\frac{j}{2}}(x),
$$

where *c <sup>j</sup>* 2 ¼ *uh x <sup>j</sup>* 2 � �≈*u x <sup>j</sup>* 2 � � are the unknowns coefficients. This formulation leads to solve in <sup>2</sup>*N*�<sup>1</sup> a linear system:

*A***c** ¼ **b**,

where **c** ¼ *c*<sup>1</sup> 2 ,*c*1, … ,*cN*�<sup>1</sup> 2 h i*<sup>t</sup>* ∈ <sup>2</sup>*N*�<sup>1</sup> is the unknown vector containing the coefficients *c <sup>j</sup>* 2 , *j* ¼ 1, 2, … , 2*N* � 1, *A* is an 2ð Þ� *N* � 1 ð Þ 2*N* � 1 matrix with entries

$$a\_{ij} = \int\_{a}^{b} (\phi\_{\frac{1}{2}}^{\prime} \phi\_{\frac{j}{2}}^{\prime} + q \phi\_{\frac{i}{2}} \phi\_{\frac{j}{2}}) d\infty, \quad i, j = 1, 2, \dots, 2N - 1,$$

and load vector **b**∈ <sup>2</sup>*N*�<sup>1</sup> has entries

$$b\_{\frac{i}{2}} = \int\_{a}^{b} f \phi\_{\frac{i}{2}} d\mathbf{x}, \quad i = 1, 2, \dots, 2N - 1.$$

Since the shape functions *ϕ<sup>i</sup>* have a small support, the matrix *A* is mostly composed of zeros. However, the main difference with the Lagrange *P*<sup>1</sup> FE method, the matrix *A* is no longer a tridiagonal matrix.

**Computer Implementation**: The coefficients of the matrix *A* can be computed more easily by considering the following change of variables, for *ξ*∈½ � �1, 1 :

$$\begin{aligned} \boldsymbol{\omega} &= \frac{\boldsymbol{\omega}\_{j} + \boldsymbol{\omega}\_{j-1}}{2} + \frac{\boldsymbol{\omega}\_{j} - \boldsymbol{\omega}\_{j-1}}{2} \boldsymbol{\xi} = \boldsymbol{\omega}\_{j-\frac{1}{2}} + \frac{\boldsymbol{\omega}\_{j} - \boldsymbol{\omega}\_{j-1}}{2} \boldsymbol{\xi}, \quad \forall \ \boldsymbol{\omega} \in [\boldsymbol{\omega}\_{j-1}, \boldsymbol{\omega}\_{j}],\\ \boldsymbol{j} &= \mathbf{1}, \boldsymbol{2}, \ldots, N. \end{aligned}$$

Hence, the shape functions can be reduced to only three basic shape functions (**Figure 3**):

$$
\hat{\phi}\_{-1}(\xi) = \frac{\xi(\xi - \mathbf{1})}{2}, \quad \hat{\phi}\_0(\xi) = (\mathbf{1} - \xi)(\mathbf{1} + \xi), \quad \hat{\phi}\_1(\xi) = \frac{\xi(\xi + \mathbf{1})}{2}.
$$

Their respective derivatives are

$$\frac{d\hat{\phi}\_{-1}(\xi)}{d\xi} = \frac{2\xi - 1}{2}, \quad \frac{d\hat{\phi}\_0(\xi)}{d\xi} = -2\xi, \quad \frac{d\hat{\phi}\_1(\xi)}{d\xi} = \frac{2\xi + 1}{2}.$$

This approach consists in considering all computations on an interval *Ii* ¼ *xi*�1, *xi* ½ � on the reference interval ½ � �1, 1 . Thus, we have:

$$\frac{d\phi\_i(\mathbf{x})}{d\mathbf{x}} = \frac{d\phi\_i(\mathbf{x}\_{i-1/2} + \frac{\mathbf{x}\_i - \mathbf{x}\_{i-1}}{2}\boldsymbol{\xi})}{d\boldsymbol{\xi}} \frac{d\boldsymbol{\xi}}{d\mathbf{x}} = \frac{2}{\boldsymbol{\chi}\_i - \boldsymbol{\chi}\_{i-1}} \frac{d\hat{\phi}\_k(\boldsymbol{\xi})}{d\boldsymbol{\xi}} = \frac{2}{h\_i} \frac{d\hat{\phi}\_k(\boldsymbol{\xi})}{d\boldsymbol{\xi}}.$$

In this case, the elementary contributions of the element *Ii* to the stiffness matrix and to the mass matrix are given by the 3 � 3 matrices *<sup>K</sup>Ii* and *<sup>M</sup>Ii* :

$$\begin{split} K^{I\_i} &= \int\_{I\_i} \begin{bmatrix} \phi\_{i-1}^{\prime}\phi\_{i-1}^{\prime} & \phi\_{i-1}^{\prime}\phi\_{i-\frac{1}{2}}^{\prime} & \phi\_{i-1}^{\prime}\phi\_{i}^{\prime} \\ \phi\_{i-\frac{1}{2}}^{\prime}\phi\_{i-1}^{\prime} & \phi\_{i-\frac{1}{2}}^{\prime}\phi\_{i-\frac{1}{2}}^{\prime} & \phi\_{i-\frac{1}{2}}^{\prime}\phi\_{i}^{\prime} \\ \phi\_{i}^{\prime}\phi\_{i-1}^{\prime} & \phi\_{i}^{\prime}\phi\_{i-\frac{1}{2}}^{\prime} & \phi\_{i}^{\prime}\phi\_{i}^{\prime} \end{bmatrix} d\mathbf{x} = \frac{2}{h\_{i}} \begin{bmatrix} \hat{\phi}\_{-1}^{\prime}\hat{\phi}\_{-1}^{\prime} & \hat{\phi}\_{-1}^{\prime}\hat{\phi}\_{0}^{\prime} & \hat{\phi}\_{-1}^{\prime}\hat{\phi}\_{1}^{\prime} \\ \hat{\phi}\_{0}^{\prime}\hat{\phi}\_{-1}^{\prime} & \hat{\phi}\_{0}^{\prime}\hat{\phi}\_{0}^{\prime} & \hat{\phi}\_{0}^{\prime}\hat{\phi}\_{1}^{\prime} \\ \hat{\phi}\_{1}^{\prime}\hat{\phi}\_{-1}^{\prime} & \hat{\phi}\_{1}^{\prime}\hat{\phi}\_{0}^{\prime} & \hat{\phi}\_{1}^{\prime}\hat{\phi}\_{1}^{\prime} \end{bmatrix} d\xi \\ = \frac{1}{3h\_{i}} \begin{bmatrix} \mathcal{T} & -8 & \mathbf{1} \\ -8 & 16 & -8 \\ \mathbf{1} & -8 & \mathbf{7} \end{bmatrix}, \end{split}$$

*A Brief Summary of the Finite Element Method for Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.95423*

$$\begin{split} M^{l\_i} &= \int\_{I\_i} \begin{bmatrix} \phi\_{i-1}\phi\_{i-1} & \phi\_{i-1}\phi\_{i-\frac{1}{2}} & \phi\_{i-1}\phi\_i\\ \phi\_{i-\frac{1}{2}}\phi\_{i-1} & \phi\_{i-\frac{1}{2}}\phi\_{i-\frac{1}{2}} & \phi\_{i-\frac{1}{2}}\phi\_i\\ \phi\_i\phi\_{i-1} & \phi\_i\phi\_{i-\frac{1}{2}} & \phi\_i\phi\_i \end{bmatrix} dx = \frac{h\_i}{2} \begin{bmatrix} \hat{\phi}\_{-1}\hat{\phi}\_{-1} & \hat{\phi}\_{-1}\hat{\phi}\_0 & \hat{\phi}\_{-1}\hat{\phi}\_1\\ \hat{\phi}\_0\hat{\phi}\_{-1} & \hat{\phi}\_0\hat{\phi}\_0 & \hat{\phi}\_0\hat{\phi}\_1\\ \hat{\phi}\_1\hat{\phi}\_{-1} & \hat{\phi}\_{-1}\hat{\phi}\_0 & \hat{\phi}\_1\hat{\phi}\_1 \end{bmatrix} d\xi \\ &= \frac{h\_i}{30} \begin{bmatrix} 4 & 2 & -1\\ 2 & 16 & 2\\ -1 & 2 & 4 \end{bmatrix}. \end{split}$$

**Coefficients of the right-hand side b**: Usually, the function *f* is only known by its values at the mesh points *xi* 2 , *i* ¼ 0, 1, … , 2*N* and thus, we use the decomposition of *f* in the basis of shape functions *ϕ<sup>i</sup>* 2 , *<sup>i</sup>* <sup>¼</sup> 0, 1, … , 2*<sup>N</sup>* as *f x*ð Þ¼ <sup>P</sup>2*<sup>N</sup> <sup>j</sup>*¼0*f x <sup>j</sup>* 2 � �*<sup>ϕ</sup> <sup>j</sup>* 2 . Each component *bi* <sup>2</sup> of the right-hand side vector is obtained as *bi* <sup>2</sup> <sup>¼</sup> <sup>P</sup>*<sup>N</sup> k*¼1 Ð *xk xk*�<sup>1</sup> *fϕ<sup>i</sup>* 2 *dx*. Using the previous decomposition of *f*, we obtain:

$$b\_{\frac{i}{2}} = \sum\_{k=1}^{N} \int\_{\mathbf{x}\_{k-1}}^{\mathbf{x}\_{k}} \sum\_{j=0}^{2N} f\left(\mathbf{x}\_{\frac{i}{2}}\right) \phi\_{\frac{j}{2}} \phi\_{\frac{i}{2}} d\mathbf{x} = \sum\_{j=0}^{2N} f\left(\mathbf{x}\_{\frac{j}{2}}\right) \left(\sum\_{k=1}^{N} \int\_{\mathbf{x}\_{k-1}}^{\mathbf{x}\_{k}} \phi\_{\frac{i}{2}} \phi\_{\frac{j}{2}} d\mathbf{x}\right).$$

Thus, the problem is reduced to computing the integrals Ð *xk xk*�<sup>1</sup> *ϕi* 2 *ϕ <sup>j</sup>* 2 *dx*. It is easy to see that we obtain expressions very similar to that of the mass matrix. More precisely, the element *Ii* ¼ *xi*�1, *xi* ½ � will contribute to only three components of indices *<sup>i</sup>* � 1, *<sup>i</sup>* � <sup>1</sup> <sup>2</sup> and *i* as:

$$\mathbf{b}^{I\_i} = \frac{h\_i}{30} \begin{bmatrix} 4 & 2 & -1 \\ 2 & 16 & 2 \\ -1 & 2 & 4 \end{bmatrix} \begin{bmatrix} f(\mathbf{x}\_{i-1}) \\ f\left(\mathbf{x}\_{i-\frac{1}{2}}\right) \\ f(\mathbf{x}\_i) \end{bmatrix}.$$

#### *2.5.2 Nonhomogeneous boundary conditions*

Consider the following two-point BVP: find *u* ∈*C*<sup>2</sup> ð Þ *a*, *b* such that

$$-u'' + q(\mathbf{x})u = f(\mathbf{x}), \quad \mathbf{x} \in [a, b], \quad u(a) = a, \quad u(b) = \beta,\tag{29}$$

where *α* and *β* are given constants and *f* ∈*C a*ð Þ , *b* is a given function.

Multiplying (29) by a function *v*∈ *H*<sup>1</sup> <sup>0</sup> ¼ *v* : k k*v* <sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>0</sup> k k<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup>, *v a*ð Þ¼ *v b*ð Þ¼ <sup>0</sup> n o and integrating by parts gives

$$\int\_{a}^{b} fv dx = \int\_{a}^{b} (-u'' + qu)v dx = -u'(b)v(b) + u'(a)v(a) + \int\_{a}^{b} (u'v' + quv) dx = \int\_{a}^{b} u'v' dx.$$

Hence, the weak or variational form of (29) reads: Given *u a*ð Þ¼ *α*, *u b*ð Þ¼ *β*, find *<sup>u</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>¼</sup> *<sup>v</sup>* : k k*<sup>v</sup>* <sup>2</sup> <sup>þ</sup> *<sup>v</sup>*<sup>0</sup> k k<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup> n o, such that

$$\int\_{a}^{b} (u'v' + quv)d\mathfrak{x} = \int\_{a}^{b} fv d\mathfrak{x}, \quad \forall \ v \in H\_0^1.$$

Let *V*<sup>2</sup> *<sup>h</sup>* and *V*<sup>2</sup> *<sup>h</sup>*,0, respectively, be the space of all continuous piecewise quadratic functions and the space of all continuous piecewise quadratic functions which

vanish at the end points *a* and *b*, on a uniform partition *a* ¼ *x*<sup>0</sup> <*x*<sup>1</sup> < ⋯ <*xN* ¼ *b* of the interval ½ � *a*, *b* .

The FE method scheme consists of finding *uh* ∈*V*<sup>2</sup> *<sup>h</sup>*, such that:

$$\int\_{a}^{b} u'\_h v' d\mathfrak{x} + \int\_{a}^{b} q u\_h v d\mathfrak{x} = \int\_{a}^{b} f v d\mathfrak{x}, \quad \forall \ v \in V\_{h,0}^2.$$

Introduce the notation *x <sup>j</sup>* 2 , *j* ¼ 0, 1, … , 2*N* � 1, 2*N* for the mesh points and *ϕ <sup>j</sup>* 2 , *<sup>j</sup>* <sup>¼</sup> 0, 1, … , 2*<sup>N</sup>* � 1, 2*<sup>N</sup>* for the basis of *<sup>V</sup>*<sup>2</sup> *<sup>h</sup>* and *ϕ <sup>j</sup>* 2 , *j* ¼ 1, … , 2*N* � 1 for the basis of *V*<sup>2</sup> 0,*h*. Using these notations, we have:

$$u\_h = \sum\_{j=0}^{2N} c\_{\frac{j}{2}} \phi\_{\frac{j}{2}}(\mathbf{x}),$$

where *c <sup>j</sup>* 2 ¼ *uh x <sup>j</sup>* 2 � �≈*u x <sup>j</sup>* 2 � � are the unknowns coefficients. We note that *<sup>c</sup>*<sup>0</sup> <sup>¼</sup> *uh*ð Þ¼ *<sup>x</sup>*<sup>0</sup> *<sup>α</sup>* and *<sup>c</sup>*2*<sup>N</sup>* <sup>¼</sup> *uh*ð Þ¼ *xN <sup>β</sup>*. This formulation leads to solve in <sup>2</sup>*N*�<sup>1</sup> a linear system:

$$\mathbf{A}\mathbf{c} = \mathbf{b},$$

where **c** ¼ *c*<sup>1</sup> 2 ,*c*1, … ,*cN*�<sup>1</sup> 2 h i*<sup>t</sup>* ∈ <sup>2</sup>*N*�<sup>1</sup> is the unknown vector containing the coefficients *c <sup>j</sup>* 2 , *j* ¼ 1, 2, … , 2*N* � 1, *A* is an 2ð Þ� *N* � 1 ð Þ 2*N* � 1 matrix with entries

$$a\_{ij} = \int\_{a}^{b} (\phi\_{\frac{i}{2}}^{\prime} \phi\_{\frac{j}{2}}^{\prime} + q \phi\_{\frac{i}{2}} \phi\_{\frac{j}{2}}) d\mathbf{x}, \quad i, j = 1, 2, \dots, 2N - 1,$$

and the load vector **b**∈ <sup>2</sup>*N*�<sup>1</sup> has entries

$$b\_{\ddagger} = \int\_{a}^{b} f \phi\_{\ddagger} d\mathbf{x} - a \int\_{a}^{b} \left(\phi\_{\ddagger}^{\prime} \phi\_{0}^{\prime} + q \phi\_{\ddagger} \phi\_{0}\right) d\mathbf{x} - \beta \int\_{a}^{b} \left(\phi\_{\ddagger}^{\prime} \phi\_{N}^{\prime} + q \phi\_{\ddagger} \phi\_{N}\right) d\mathbf{x}, \quad i = 1, 2, \dots, 2N - 1.$$

Clearly, the only extra terms are given in the vector with entries

$$\tilde{b}\_{\frac{i}{2}} = -a \int\_{a}^{b} \left(\phi\_{\frac{i}{2}}^{\prime} \phi\_{0}^{\prime} + q \phi\_{\frac{i}{2}} \phi\_{0} \right) d\mathbf{x} - \beta \int\_{a}^{b} \left(\phi\_{\frac{i}{2}}^{\prime} \phi\_{N}^{\prime} + q \phi\_{\frac{i}{2}} \phi\_{N} \right) d\mathbf{x}, \quad i = 1, 2, \dots, 2N - 1.$$

Suppose *q* ¼ 0 then for *N* ≥2, we have

~ *b*1 <sup>2</sup> ¼ �*α* ð*b a ϕ*0 1 2 *ϕ*0 <sup>0</sup>*dx* � *β* ð*b a ϕ*0 1 2 *ϕ*0 *Ndx* ¼ �*α* ð*<sup>x</sup>*<sup>1</sup> *x*0 *ϕ*0 1 2 *ϕ*0 <sup>0</sup> <sup>¼</sup> <sup>8</sup>*<sup>α</sup>* 3*h*<sup>1</sup> , ~ *b*<sup>1</sup> ¼ �*α* ð*b a ϕ*0 1*ϕ*0 <sup>0</sup>*dx* � *β* ð*b a ϕ*0 1*ϕ*0 *Ndx* ¼ �*α* ð*<sup>x</sup>*<sup>1</sup> *x*0 *ϕ*0 1*ϕ*0 <sup>0</sup> ¼ � *<sup>α</sup>* 3*h*<sup>1</sup> , ~ *bi* <sup>2</sup> ¼ �*α* ð*b a ϕ*0 *i* 2 *ϕ*0 <sup>0</sup>*dx* � *β* ð*b a ϕ*0 *i* 2 *ϕ*0 *Ndx* ¼ 0, *i* ¼ 3, … , 2*N* � 3, ~ *bN*�<sup>1</sup> ¼ �*α* ð*b a ϕ*0 *<sup>N</sup>*�<sup>1</sup>*ϕ*<sup>0</sup> <sup>0</sup>*dx* � *β* ð*b a ϕ*0 *<sup>N</sup>*�<sup>1</sup>*ϕ*<sup>0</sup> *Ndx* ¼ �*β* ð*xN xN*�<sup>1</sup> *ϕ*0 *<sup>N</sup>*�<sup>1</sup>*ϕ*<sup>0</sup> *Ndx* ¼ � *<sup>β</sup>* 3*h*<sup>1</sup> , ~ *bN*�<sup>1</sup> <sup>2</sup> ¼ �*α* ð*b a ϕ*0 *<sup>N</sup>*�<sup>1</sup> 2 *ϕ*0 <sup>0</sup>*dx* � *β* ð*b a ϕ*0 *<sup>N</sup>*�<sup>1</sup> 2 *ϕ*0 *Ndx* ¼ �*β* ð*xN xN*�<sup>1</sup> *ϕ*0 *<sup>N</sup>*�<sup>1</sup> 2 *ϕ*0 *Ndx* <sup>¼</sup> <sup>8</sup>*<sup>β</sup>* 3*h*<sup>1</sup> *:*

*A Brief Summary of the Finite Element Method for Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.95423*
