## 3 Finite Element Method
### The \({}_{11}t\) Local Problem: Weak Formulation  
The system of differential equations that defines the problem \({}_{11}T\) is given by  
\[\left(\beta_{11}-C_{1111}\ _{11}R_{1,1}-C_{1122}\ _{11}R_{2,2} \right)_{1}-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{2}=0,\] \[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]  
The perfect contact conditions given by Equation (14), which describe problem \({}_{11}T\), are expressed as follows  
\[||(\beta_{11}-C_{1111}\ _{11}R_{1,1}-C_{1122}\ _{11}R_{2,2}) \mathrm{n}_{1}+(-C_{1212}\ _{11}R_{1,2}-C_{1212}\ _{11}R_{2,1})\mathrm{n}_{2}||_{ \Gamma}=0,\] \[||(-C_{1212}\ _{11}R_{1,2}-C_{1212}\ _{11}R_{2,1})\mathrm{n}_{1}+ (\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2})\mathrm{n}_{2}||_{ \Gamma}=0. \tag{34}\]  
Equation (34) can also be written as  
\[||(\beta_{11}-\mathrm{Y}_{11})\mathrm{n}_{1}-\mathrm{Y}_{12} \mathrm{n}_{2}||=0,\] \[||-\mathrm{Y}_{12}\mathrm{n}_{1}+(\beta_{22}-\mathrm{Y}_{22}) \mathrm{n}_{2}||=0, \tag{35}\]  
where  
\[\mathrm{Y}_{11}=C_{1111}\frac{\partial\ _{11}R_{1}}{\partial y_{1}}+C_{1 122}\frac{\partial\ _{11}R_{2}}{\partial y_{2}},\quad\mathrm{Y}_{22}=C_{1122}\frac{ \partial\ _{11}R_{1}}{\partial y_{1}}+C_{2222}\frac{\partial\ _{11}R_{2}}{\partial y_{2}},\] \[\mathrm{Y}_{12}=C_{1212}\bigg{(}\frac{\partial\ _{11}R_{1}}{ \partial y_{2}}+\frac{\partial\ _{11}R_{2}}{\partial y_{1}}\bigg{)},\]and \(||g||\) denotes the discontinuity of the function \(g\) across the interface. From Equation (35), the following is obtained  
\[||Y_{11}n_{1}+Y_{12}n_{2}||=||\beta_{11}n_{1}||,\] \[||Y_{12}n_{1}+Y_{22}n_{2}||=||\beta_{22}n_{2}||. \tag{36}\]  
The discontinuity at the interface \(\Gamma\) is defined by  
\[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]0 \[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]1  
where the superscripts \(m\) and \(f\) are used for the matrix and fiber coefficients, respectively. Additionally, using symmetry and periodicity conditions, the obtained boundary conditions for problem \({}_{11}T\) are given by  
\[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]2 \[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]3 \[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]4 \[-\left(C_{1212}\ _{11}R_{1,2}+C_{1212}\ _{11}R_{2,1}\right)_{1}+ \left(\beta_{22}-C_{1122}\ _{11}R_{1,1}-C_{2222}\ _{11}R_{2,2}\right)_{2}=0. \tag{33}\]5  
where \(\partial\Omega_{L}\), \(\partial\Omega_{R}\), \(\partial\Omega_{U}\) and \(\partial\Omega_{D}\) represent the left, right, upper and lower boundaries of the cell quarter. Suppose that we look for the approximate solution \(\tilde{\mathbf{R}}=\left[\tilde{R}_{1}\ \ \ \tilde{R}_{2}\right]^{t}\approx R\) of the problem (33) within certain space of functions \(\mathcal{H}\) with dimension \(n\) such that  
\[\tilde{\mathbf{R}}=\left[ \begin{array}{c}\tilde{R}_{1}\\ \tilde{R}_{2}\end{array} \right]=\sum_{i=1}^{n} \begin{bmatrix}N_{i}&0\\ 0&N_{i}\end{bmatrix} \left[ \begin{array}{c}q_{1i}\\ q_{2i}\end{array} \right]=\mathbf{N}\mathbf{q}, \tag{37}\]  
where \(\{N_{1}\ N_{2}\ \cdots\ N_{n}\}\) is a basis of the space \(\mathcal{H}\) and  
\[\mathbf{N}= \begin{bmatrix}N_{1}&0&N_{2}&0&\cdots&N_{n}&0\\ 0&N_{1}&0&N_{2}&\cdots&0&N_{n}\end{bmatrix} .\]  
Vector \(\mathbf{q}=\left[q_{11}\ \ \ q_{21}\ \ \ q_{12}\ \ \ q_{22}\ \ \cdots\ \ q_{1n}\ \ \ q_{2n}\right]^{t}\) represents the nodal values of the unknown field.