The Laplace equation in polar coordinate system is expressed as

where the radial coordinate is denoted by

$r$ and the angular coordinate is denoted by

$\theta $. The solution of the Laplace governing equation is approximated by using the Trefftz basis functions satisfying the governing equation, as shown in Equation (10). The Trefftz basis functions are obtained by finding the general solutions using the separation of variables method [

35]. The Trefftz basis functions can be found to solve problems in a simply connected domain, as shown in

Figure 2.

#### 3.1. Formulation of T-Complete Basis Functions

We may apply the separation of variables [

36]. The solution may be in the following form:

Then, Equation (10) can be rewritten as follows.

We divide

$U(r)V(\theta )$ on both sides in the above equation and the equation can be rewritten as two differential equations as follows.

Using the constant

$v$ to ensure positive or negative constants, we have

$\lambda =0$,

$\lambda ={v}^{2}$ and

$\lambda =-{v}^{2}$. Considering the first scenario

$\lambda =0$, we obtain the solutions as follows.

where

${D}_{1}$,

${D}_{2}$,

${D}_{3}$ and

${D}_{4}$ are constants. Using the boundary conditions of

$V(r,0)=V(r,2\pi )$, we may find that

${D}_{1}=0$. Substituting Equations (16) and (17) into Equation (11), we have

where

${a}_{0}$ and

${b}_{0}$ denote the coefficients. Considering the second scenario,

$\lambda ={v}^{2}$, we obtain the following solutions.

where

${D}_{5}$,

${D}_{6}$,

${D}_{7}$ and

${D}_{8}$ are the coefficients. Inserting the above equations into Equation (11), we obtain

where

$a$,

$b$,

$c$ and

$d$ denote the coefficients. Then, we may consider the last scenario

$\lambda =-{v}^{2}$. Since there is not able to find any non–zero periodic solutions of differential system for

$U(r)$, we may only find

$V(\theta )=0$. Collecting all the solutions from the above results, the linearly independent solutions to Laplace equation can be obtained as follows.

The Trefftz basis functions in a simply connected domain are as follows.

In the numerical analysis, we approach the general solution in the form of infinite series of the Laplace equation in a simply connected domain by using a finite number of

$m$. As a result, Equation (23) can be rewritten as

where

$m$ represents the terms of the Trefftz order. The above Equation (24) can be used to match the Dirichlet boundary condition. We may also need to consider the Neumann boundary conditions as follows.

Equation (25) can be rewritten as

where

$\nabla $ is the gradient and

$\overrightarrow{n}=({n}_{x},{n}_{y})$ denotes the normal vector. Equation (25) can then be written as

where

Using Equation (24), we may find the derivatives of

$\partial \phi /\partial r$ and

$\partial \phi /\partial \theta $ as follows.

Using Equations (29) and (30), Equation (27) leads to

#### 3.2. The Characteristic Length

The characteristic length plays a crucial role in controlling the proposed numerical approach in a stable way. Because the matrix assembled with Trefftz trial functions is a full matrix, the resultant system of linear equations may be ill–posed [

17,

18]. The accuracy of the results from the Trefftz method depends sensitively on the order of the T-complete basis functions. Besides, the numerical solution may be unstable. Related to the CTM for solving two-dimensional Laplacian problems, Liu [

17] proposed a characteristic length to mitigate the problems of the ill-posedness for the system of linear equations. Applying Dirichlet boundary condition, we obtain

Using the CTM, we obtain the approximation solution of the Laplace equation as follows.

where

${\mathbf{b}}_{v}=[\begin{array}{ccc}{a}_{0}& {a}_{v}& {c}_{v}\end{array}]$,

${\mathbf{T}}_{v}=[\begin{array}{ccc}1& {(r/R)}^{v}\mathrm{cos}(v\theta )& {(r/R)}^{v}\mathrm{sin}(v\theta )\end{array}]$,

$\mathbf{x}$ is the coordinate of the collocation points and

$\mathbf{x}\in \mathsf{\Omega}$. Applying the Neumann boundary condition, we may obtain the following equations for simply connected domain using the characteristic length.

To mitigate the ill-posedness, the characteristic length [

19],

$R$, is adopted and is expressed as

where

$maximum(r)$ denotes the maximum radial distance in the problem domain. After adopting the characteristic length in our numerical model, the ill-posed phenomenon is greatly reduced, and the accurate numerical solutions can be obtained. Collocating the numerical expansion from Equations (32) and (34) at boundary collocation points to match the given boundary conditions, we may obtain the following equation.

where

$\mathbf{T}$ represents a

$l\times M$ matrix,

$M=2m+1$,

$\mathbf{b}$ represents a

$M\times 1$ vector of unknown coefficients,

$\mathbf{B}$ denotes a vector (size of

$l\times 1$) of given functions at boundary points,

$l$ represents the number of the boundary points and

$M$ represents the terms of the Trefftz order.

$i\le l$ and

$j\le l$ in which

$i$ and

$j$ are the number of boundary points for Dirichlet and Neumann boundary conditions, respectively.

${g}_{1},{g}_{2},\dots ,{g}_{i}$ and

${f}_{1},{f}_{2},\dots ,{f}_{j}$ denote the boundary values for Dirichlet and Neumann boundary conditions, respectively.

In this article, we adopt the domain decomposition method (DDM) [

37,

38] to solve the nonlinear moving boundary problems in heterogeneous geological media. The DDM is commonly used to solve the problem with different physical characteristics in each subdomain. We first split the domain into two subdomains which are intersected only at the interface. Hence, each subdomain can be regarded as an independent soil layer with its own hydraulic conductivity. At the interface, the flux and the head must satisfy the continuity condition. For instance, we consider a rectangular domain,

$\mathsf{\Omega}$, which can be split into two intersected subdomains,

${\mathsf{\Omega}}_{1}$ and

${\mathsf{\Omega}}_{2}$.

Figure 3 shows that the rectangular domain is divided into

${\mathsf{\Gamma}}_{1}$,

${\mathsf{\Gamma}}_{2}$, …,

${\mathsf{\Gamma}}_{8}$ sub boundaries where

${\mathsf{\Gamma}}_{1}$,

${\mathsf{\Gamma}}_{2}$, …,

${\mathsf{\Gamma}}_{4}$ and

${\mathsf{\Gamma}}_{5}$,

${\mathsf{\Gamma}}_{6}$, …,

${\mathsf{\Gamma}}_{8}$ are sub boundaries of subdomains Ω

_{1}. and

${\mathsf{\Omega}}_{2}$, respectively. At the interface, the sub boundaries,

${\mathsf{\Gamma}}_{2}$ and

${\mathsf{\Gamma}}_{6}$, are overlapped at the same location. Therefore, additional boundary conditions are imposed on the boundary points to ensure the flux and the head at the interface must be the same.

Matching all given boundary conditions, we may obtain a system of linear equations as

where

${\mathbf{T}}_{{\mathsf{\Omega}}_{1}}$ with the size of

${l}_{1}\times {M}_{1}$ and

${\mathbf{T}}_{{\mathsf{\Omega}}_{2}}$ with the size of

${l}_{2}\times {M}_{2}$ are the

$\mathbf{T}$ matrix shown in Equation (37) for

${\mathsf{\Omega}}_{1}$ and

${\mathsf{\Omega}}_{2}$, respectively.

${l}_{1}$ and

${l}_{2}$ are the number of boundary points;

${M}_{1}$ and

${M}_{2}$ are the T-complete basis function order for

${\mathsf{\Omega}}_{1}$ and

${\mathsf{\Omega}}_{2}$, respectively.

${{\mathbf{T}}_{\mathrm{I}}|}_{{\mathsf{\Gamma}}_{2}}$ of the boundary

${\mathsf{\Gamma}}_{2}$ with the size of

${l}_{I}\times {M}_{1}$ and

${{\mathbf{T}}_{\mathrm{I}}|}_{{\mathsf{\Gamma}}_{6}}$ of the boundary

${\mathsf{\Gamma}}_{6}$ with the size of

${l}_{I}\times {M}_{2}$ are matrices at the interface.

${l}_{I}$ represents the boundary point number at the interface,

${\mathbf{0}}_{{\mathsf{\Omega}}_{1}}$ and

${\mathbf{0}}_{{\mathsf{\Omega}}_{2}}$ are matrices which all values are zero with the size of

${l}_{2}\times {M}_{1}$ and

${l}_{1}\times {M}_{2}$, respectively.

${\mathbf{b}}_{{\mathsf{\Omega}}_{1}}$ denotes a

${M}_{1}\times 1$ vector of unknown coefficients of

${\mathsf{\Omega}}_{1}$,

${\mathbf{b}}_{{\mathsf{\Omega}}_{2}}$ denotes a

${M}_{2}\times 1$ vector of unknown coefficients of

${\mathsf{\Omega}}_{2}$.

${\mathbf{B}}_{{\mathsf{\Omega}}_{1}}$ and

${\mathbf{B}}_{{\mathsf{\Omega}}_{2}}$ denote vectors of given functions at boundary points of

${\mathsf{\Omega}}_{1}$ and

${\mathsf{\Omega}}_{2}$, respectively.

${B}_{I}=\begin{array}{cc}[{\mathbf{0}}_{\mathrm{g}}& {\mathbf{0}}_{\mathrm{f}}\end{array}{]}^{\mathrm{T}}$,

${\mathbf{0}}_{\mathrm{g}}$ and

${\mathbf{0}}_{\mathrm{f}}$ are vectors which all values are zero with the size of

${l}_{\mathrm{I}}\times 1$. The total head can be determined by collocating the inner points within subdomains,

${\mathsf{\Omega}}_{1}$ and

${\mathsf{\Omega}}_{2}$. Consequently, the value of the total head,

$\phi $, can then be approximated by using Equation (33).