## 2 Radial Point Interpolation Method
It can be further assumed that only the nodes located within the influence domain of a specific point of interest, \(\mathbf{x}_{I}\), have an impact on the value of the function, \(\mathbf{u}(\mathbf{x}_{I})\), at that point:  
\[u(\mathbf{x}_{I})=\sum_{i=1}^{n}R_{i}(\mathbf{x}_{I})a_{i}(\mathbf{x}_{I})\ + \sum_{j=1}^{m}p_{j}(\mathbf{x}_{I})b_{j}(\mathbf{x}_{I})=\left\{\mathbf{R}^{ \mathbf{T}}(\mathbf{x}_{I}),\mathbf{p}^{\mathbf{T}}(\mathbf{x}_{I})\right\} \left\{ \begin{array}{l}\mathbf{a}\\ \mathbf{b}\end{array} \right\}, \tag{1}\]  
where \(R_{i}(\mathbf{x}_{I})\) represents a radial basis function (RBF), \(n\) and \(m\) stand for the number of nodes within the influence domain of \(\mathbf{x}_{I}\) and the number of monomials of the polynomial basis function, respectively, and \(a_{i}(\mathbf{x}_{I})\) and \(b_{j}(\mathbf{x}_{I})\) represent non-constant coefficients of \(R_{i}(\mathbf{x}_{I})\) and \(p_{j}(\mathbf{x}_{I})\), respectively. Several radial basis functions are available in the literature [7], and many have been explored and developed over time. This work will utilise the multiquadratics (MQ) RBF, first introduced by [PERSON] [23]. This function is widely recognised as yielding superior results and is often paired with RPI [7]. The MQ RBF can be described as follows:  
\[R_{i}(\mathbf{x}_{I})=R(r_{li})=(r_{li}^{2}+c^{2})^{p} \tag{2}\]  
for which,  
\[r_{li}=\sqrt{(x_{i}-x_{I})^{2}+(y_{i}-y_{I})^{2}}. \tag{3}\]  
The literature shows that shape parameters \(c\) and \(p\) considerably influence the shape of the final interpolation function and therefore the accuracy of the final results [7; 17]. Therefore, following the literature recommendation, in this work, the following shape parameters are assumed: \(c=1.42\) and \(p=1.03\)[7; 17]. These values will allow us to ensure that the constructed shape functions actually possess the delta Kronecker property. To ensure a unique approximation, the following additional system of equations is introduced:  
\[\sum_{i=1}^{n}p_{j}(x_{i})a_{i}(x_{i})=0,\ \ j=1,2,...,m. \tag{4}\]  
With Equations (1) and (4), it is possible to write  
\[\left\{ \begin{bmatrix}u_{1}\\ u_{2}\\ \vdots\\ u_{n}\\ 0\\ 0\\ 0\end{bmatrix} \right\}=\left[\begin{bmatrix}R(r_{11})&R(r_{12})&\cdots&R(r_{1n })\\ R(r_{21})&R(r_{22})&\cdots&R(r_{2n})\\ \vdots&\vdots&&\vdots\\ R(r_{n1})&R(r_{n2})&\cdots&R(r_{m})\\ \begin{bmatrix}1&1&\cdots&1\\ x_{1}&x_{2}&\cdots&x_{n}\\ y_{1}&y_{2}&\cdots&y_{n}\end{bmatrix} & \begin{bmatrix}1&x_{1}&y_{1}\\ 1&x_{2}&y_{2}\\ \vdots&\vdots&\vdots\\ 1&x_{n}&y_{n}\\ \end{bmatrix} \\ \begin{bmatrix}1&1&\cdots&1\\ x_{1}&x_{2}&\cdots&x_{n}\\ y_{1}&y_{2}&\cdots&y_{n}\end{bmatrix} & \begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix} \right]\left\{ \begin{bmatrix}a_{1}\\ a_{2}\\ \vdots\\ a_{n}\\ b_{1}\\ b_{2}\\ b_{3}\end{bmatrix} \right\}, \tag{5}\]  
which allows us to obtain the non-constant coefficients \(a_{i}(\mathbf{x}_{I})\) and \(b_{j}(\mathbf{x}_{I})\):  
\[\left\{ \begin{bmatrix}u_{1}\\ u_{2}\\ u_{n}\\ u_{n}\\ 0\\ 0\end{bmatrix} \right\}=\mathbf{G}\left\{ \begin{bmatrix}a_{1}\\ a_{2}\\ \vdots\\ a_{n}\\ b_{1}\\ b_{2}\\ b_{3}\end{bmatrix} \right\}\Longrightarrow\left\{ \begin{bmatrix}a_{1}\\ a_{2}\\ \vdots\\ a_{n}\\ b_{1}\\ b_{2}\\ b_{3}\end{bmatrix} \right\}=\mathbf{G}^{-1}\left\{ \begin{bmatrix}u_{1}\\ u_{2}\\ \vdots\\ u_{n}\\ 0\\ 0\end{bmatrix} \right\}=\mathbf{G}^{-1}\left\{ \begin{bmatrix}u_{s}\\ 0\\ 0\\ 0\end{bmatrix} \right\}. \tag{6}\]  
Incorporating \(a_{i}(\mathbf{x}_{I})\) and \(b_{j}(\mathbf{x}_{I})\) into Equation (1), the following interpolation is achieved:\[\mathbf{u}(\mathbf{x}_{I})=\Big{\{}\mathbf{R}^{T}(\mathbf{x}_{I}),\mathbf{p}^{T}( \mathbf{x}_{I})\Big{\}}\mathbf{G}^{-1}\Bigg{\{} \begin{matrix}\mathbf{u}_{ \mathbf{s}}\\ \mathbf{0}\end{matrix}