## 2 The Steady-State Allen-Cahn Equation and a Modified Polynomial Expansion Method  
We start with the steady-state Allen-Cahn equation as follows:  
\[\Delta u(x,y)=u-u^{3}+H(x,y),(x,y)\in\Omega, \tag{1}\]  
\[u(x,y)=G(x,y),(x,y)\in\Gamma_{1}, \tag{2}\]\[u_{n}(x,y)=F(x,y),(x,y)\in\Gamma_{2}, \tag{3}\]  
where \(\Delta\) is the Laplacian operator, and \(F\), \(G\) and \(H\) are given functions. \(\Gamma\) is the boundary of issue domain \(\Omega\) with \(\Gamma=\Gamma_{1}\cup\Gamma_{2}\), and \(n\) is a unit outward normal on \(\Gamma\). Under the presented Dirichlet boundary condition (2) and the Neumann boundary condition (3), we can resolve Equation (1) to acquire a solution of \(u(x,y)\). Then, we employ the polynomial expansion as a trial solution of the PDE and formulate the needed algebraic equations after a suitable collocation in the problem area. However, this is rarely utilized as a major numerical tool to resolve the nonlinear PDEs. The main reason being that the resultant nonlinear algebraic equations (NAEs) are often seriously ill-conditioned. The elements in the polynomial matrix are as follows:  
\[\left[ \begin{array}{cccccccc}1&y&y^{2}&\cdots&y^{m-1}&y^{m}\\ x&xy&xy^{2}&\cdots&xy^{m-1}&xy^{m}\\ x^{2}&x^{2}y&x^{2}y^{2}&\cdots&x^{2}y^{m-1}&x^{2}y^{m}\\ \vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\ x^{m}&x^{m}y&x^{m}y^{2}&\cdots&x^{m}y^{m-1}&x^{m}y^{m}\end{array} \right] \tag{4}\]  
are often utilized to expand the solution of \(u(x,y)\). If the elements are restrained from the upper-left triangle, the result is the famous Pascal triangle expansion:  
\[ \begin{array}{cccccccc}&&1&&&&\\ &&x&y&&\\ &&x^{2}&xy&y^{2}&&\\ &&x^{3}&x^{2}y&xy^{2}&y^{3}&&\\ &&x^{4}&x^{3}y&x^{2}y^{2}&xy^{3}&y^{4}&&\\ x^{5}&x^{4}y&x^{3}y^{2}&x^{2}y^{3}&xy^{4}&y^{5}&\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\end{array}  \tag{5}\]  
Consequently, the solution \(u(x,y)\) is expanded by  
\[u(x,y)=\sum_{i=1}^{m}\sum_{j-1}^{i}p_{ij}x^{i-j}y^{j-1}, \tag{6}\]  
where the coefficients \(p_{\bar{y}}\) are to be resolved, where the number of all elements is \(n=m(m+1)/2\). The highest order of the above polynomial is \(m-1\). From Equation (6), the following equations can be directly stated:  
\[u_{x}(x,y)=\sum_{i=1}^{m}\sum_{j-1}^{i}p_{ij}(i-j)x^{i-j-1}y^{j-1}, \tag{7}\]  
\[u_{y}(x,y)=\sum_{i=1}^{m}\sum_{j-1}^{i}p_{ij}(j-1)x^{i-j}y^{j-2}, \tag{8}\]  
\[u_{n}\Big{|}_{(x,y)\in\Gamma}=\sum_{i=1}^{m}\sum_{j-1}^{i}p_{ij}\Big{[}(i-j)x^ {i-j-1}y^{j-1}n_{x}+(j-1)x^{i-j}y^{j-2}n_{y}\Big{]}, \tag{9}\]  
\[\Delta u(x,y)=\sum_{i=1}^{m}\sum_{j-1}^{i}p_{i\bar{y}}[(i-j)(i-j-1)x^{i-j-2}y^ {j-1}+(j-1)(j-2)x^{i-j}y^{j-3}]. \tag{10}\]  
Introducing these equations to Equations (1)-(3), and selecting \(n_{1}\) and \(n_{2}\) collocation points on the boundary and in the area, to satisfy the boundary condition and the field equation, we obtain a system of NAEs to deal with the \(n\) coefficients \(p_{\bar{y}}\). The multi-scale Pascal triangle is then introduced as follows. Because \(x\) and \(y\) in the issue area \(\Omega\) may be arbitrarily large, the above expansion would lead to a divergence of the powers \(x^{m}\) and \(y^{m}\). To acquire an accurate solution of an AC equation employing the modified Pascal triangle polynomial expansion scheme, we have to develop a more accurate and effective method to cope with these NAEs by reducing the condition numbers. A new multiple-scale Pascal triangle is obtained by expansion of \(u(x,y)\) by  
\[u(x,y)=\sum_{i=1}^{m}\sum_{j-1}^{i}p_{ij}s_{ij}x^{i-j}y^{j-1}, \tag{11}\]  
in which the scales \(s_{ij}\) are decided as below. The coefficients \(p_{ij}\) utilized in the expansion (6) can be demonstrated as an \(n\)-dimensional vector \(\mathbf{p}\) with components \(p_{k}\), \(k=1\),..., \(n\) by  
\[ \begin{array}{l}k=0\\ \text{Do }i=1\text{, }m\\ \text{Do }j=1\text{, }i\\ k=k+1\\ p_{k}=p_{\bar{y}}\\ \text{Enddo.}\end{array}