# Efficient Discretization of the Laplacian: Application to Moving Boundary Problems
## 2 Materials and Methods
### Application to the Numerical Solution of the Stefan Problem
#### 2.4.1 General Setting  
Stefan problems have been used as a mathematical model of different physical phenomena. In a number of applications, this describes the interaction of different phases or materials. This can be a dissolution of a chemical compound into an alloy [18], the solidification of a binary alloy [21], a crystallization process [22] or just the melting and freezing of ice [23].  
Taking a minimal model, the unknown function \(u:\Omega\to\mathbb{R}\) in the Stefan problem corresponds to the temperature in a melting system [24].  
At each time \(t\in[0,T]\), the domain \(\Omega\) is separated into the two disjoint ones  
\[\Omega_{0,t}=\left\{\mathbf{x}\in\Omega:u(t,\mathbf{x})<0\right\}\text{ and }\Omega_{1,t}=\left\{\mathbf{x}\in\Omega:u(t,\mathbf{x})>0\right\},\]  
so that \(\overline{\Omega}=\overline{\Omega_{0,t}}\cup\overline{\Omega_{1,t}}\). Here, \(\Omega_{0,t}\) represents the domain filled with ice, while \(\Omega_{1,t}\) represents the domain filled with water. Accordingly, the set  
\[\Gamma_{t}=\left\{\mathbf{x}=\left(x(t),y(t)\right)\in\overline{\Omega}:u(t, \mathbf{x})=0\right\}\]  
gives their common boundary.  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \(N_{x}\) & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 & 55 \\ \(N_{y}\) & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 & 40 & 44 \\ time (ms) & 22 & 20 & 27 & 41 & 55 & 73 & 95 & 119 & 146 & 174 \\ error \(\cdot 10^{3}\) & 10.9 & 5.2 & 3.1 & 2.1 & 1.6 & 1.2 & 1 & 0.87 & 0.77 & 0.7 \\ \hline \end{tabular}
\end{table}
Table 1: Simulation time (ms) and discrete \(L_{2}\)-norm error for various numbers of horizontal (\(N_{x}\)) and vertical (\(N_{y}\)) grid points in the discretization of the model problem (10).  
Figure 3: Spatial discretization using structured grids in two cases. Number of uniformly distributed grid points: 20 (**left**) and 40 (**right**) in the horizontal direction. Number of grid points: 16 (**left**) and 33 (**right**) in the vertical direction.  
We also introduce the notation  
\[u_{0}=u|_{[0,T]\times\Omega_{0,t}}\text{ and }u_{1}=u|_{[0,T]\times\Omega_{1,t}},\]  
for the solutions on the subdomains above, where, for the sake of simplicity, we omit the time parameter \(t\). With these, the governing equations can be given as  
\[\begin{cases}\partial_{t}u_{0}(t,\mathbf{x})=D_{0}\cdot\Delta u_{0}(t,\mathbf{ x}),&\mathbf{x}\in\Omega_{0,t}\\ \partial_{t}u_{1}(t,\mathbf{x})=D_{1}\cdot\Delta u_{1}(t,\mathbf{x}),& \mathbf{x}\in\Omega_{1,t}.\end{cases} \tag{11}\]  
Here, \(D_{0}\) and \(D_{1}\) are the diffusion coefficients in the corresponding domains, where usually \(D_{0}>D_{1}\). The evolution of the common boundary is given by  
\[\partial_{t}\Gamma_{t}(\mathbf{x})=-L_{H}\cdot[[\partial_{\mathbf{v}}u(t,x(t),y(t))]], \tag{12}\]  
where \(\Gamma_{t}\) is considered as a level set parametrized with \(t\) and \(L_{H}\) denotes the latent heat of solidification [25]. Physically, this gives the speed of the interface between the ice and liquid water. The symbol \([[\cdot]]\) is for the jump operator, which is defined on the common boundary \(\Gamma_{t}\) with  
\[[[\partial_{\mathbf{v}}u(t,\mathbf{x})]]=D_{0}\cdot\mathbf{\
u}_{0}(\mathbf{x}) \cdot\
abla u_{0}(t,\mathbf{x})+D_{1}\cdot\mathbf{\
u}_{1}(\mathbf{x})\cdot \
abla u_{1}(t,\mathbf{x}), \tag{13}\]  
where \(\mathbf{\
u}_{0}\) and \(\mathbf{\
u}_{1}\) denote the corresponding (opposite) outward normals.  
Note again that depending on the model, the unknown could be the concentration of some chemical compound or the degree of crystallinity. Also, in a more involved model, multiple phases (even for ice [23]) or a separate nucleation process [22] may occur.