# Efficient Discretization of the Laplacian: Application to Moving Boundary Problems
## 3 Discussion and Conclusions  
An optimization-based FD discretization of the two-dimensional Laplace operator was developed. The core of the optimization procedure was the least-square solution of an overdetermined system, where a special weighting of the corresponding equations was utilized. The pointwise execution of this procedure provides a vectorized alternative to the conventional assembly procedure. The use and benefits of this algorithm are demonstrated on the Stefan problem, which is a mathematical model of the dynamics of two-phase materials with a moving interface. With this, the computational domains also evolve with time. Consequently, at each time step of the numerical simulations, a new spatial discretization must be performed, making the speed of this process critical to the overall efficiency of the numerical method. The proposed algorithm is notably fast, relying solely on neighboring relationships between grid points. Furthermore, it is effective on non-uniform and non-rectangular grids, combining the strengths of traditional finite difference and finite element methods. The primary challenge in such simulations is that the interface often becomes unstable. This instability can lead to spurious oscillations on the free surface, \(\Gamma_{t}\), which may grow over time and ultimately disrupt the simulation. To mitigate this issue, several stabilization techniques have been proposed, see [18]. To validate our approach in this context, the choice of a flat, common interface proves to be highly suitable. Also, the algorithm proposed here is rather quick and uses only neighboring relations between the grid points. At the same time, it also works on non-uniform and non-rectangular grids, merging the advances of classical finite difference and finite element methods. Note that using classical finite element on such grids implies only first-order convergence [27]. The main application field of this new algorithm is the family of problems with moving boundaries or interfaces, including the more complex Stefan problems mentioned at the beginning of Section 2.4.1. Another very important family of such problems arises by modeling free-surface water waves. As the waves evolve, the domain filled with water will change step-by-step. When forecasting tsunamis [28] in real time, it is highly important to reduce the computing time to an affordable level for these 3-dimensional problems. The disadvantage of the method is that the optimization procedure highly depends on the weighting, which was applied for the harmonic polynomials to obtain the optimal coefficients. Another related problem is that in the proposed algorithm, we cannot guarantee a fixed spatial convergence (or consistency) order. For irregular geometries, even the optimal coefficients will not deliver a very accurate approximation of the differential operator. A natural extension of this work is to apply the method for approximating the surface Laplacian (or Laplace-Beltrami operator), a significant topic in current research. Also, the principle in Section 2.2 is applicable for any non-linear differential operator \(D\). For this, we have to modify the overdetermined system in the optimization step, where on the left-hand side of (7), we have to apply \(D\). **Author Contributions:** Conceptualization, F.I. ; methodology, F.I. ; software, S.-J.C.; validation, S.-J.C.; formal analysis, F.I. ; investigation, S.-J.C.; resources, S.-J.C.; data curation, S.-J.C.; writing--original draft preparation, F.I. ; writing--review and editing, S.-J.C.; visualization, S.-J.C.; supervision, F.I. ; project administration, F.I. ; funding acquisition, F.I. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Research, Development and Innovation Office within the framework of the Thematic Excellence Program 2021--National Research Sub programme: \"Artificial intelligence, large networks, data security: mathematical foundation and applications\".