# Efficient Discretization of the Laplacian: Application to Moving Boundary Problems
## 2 Materials and Methods
### Problem Statement and Mathematical Background  
The Laplacian of a function \(u\in C^{2}(\Omega)\) will be approximated in \(\mathbf{x}_{0}\) based on the corresponding function values in its neighboring grid points. Our structured mesh points can be recognized as the vertices of a generic quadrilateral mesh. Using such a mesh, one discretize \(\overline{\Omega}\) and a number of software tools are available to perform the approximation. For a classical survey on these, we refer to [19]. Here, the neighbors \(\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}\) and \(\mathbf{x}_{4}\) of \(\mathbf{x}_{0}\) are determined by the edges of all quadrilaterals containing \(\mathbf{x}_{0}\). Similarly, \(\mathbf{x}_{5},\mathbf{x}_{6},\mathbf{x}_{7}\) and \(\mathbf{x}_{8}\) are determined by the diagonals containing \(\mathbf{x}_{0}\). For a visualization of these, see Figure 1. At the same time, we do not need the entire structure of such a mesh, including the relation of quadrilaterals, edges and vertices. We note that within the framework of a classical finite element method, we have to perform the following steps. 1. In the domain, we have to define a mesh, numbering its elements, construct a whole mesh structure, numbering the elements (now the quadrangles), and generate the corresponding adjacency matrices:  
Figure 1: A uniform (**left**) and a general (**right**) quadrilateral grid for finite difference approximations. A uniform rectangular grid (**left**) related to the approximation (4) and a non-uniform quadrilateral grid (**right**) related to (5). 1. for each node, the number of each subdomain (element) containing this node; 2. for each element, the list of its nodes; 3. for each element, the list of its neighboring elements. 2. By mapping a reference element to each of the physical ones, one has to evaluate the element stiffness matrices, computing Jacobians for this mapping. 3. In the assembling procedure, applying a loop, for each node, we have to collect the above contributions from the adjacent elements (given by the matrix in (a)). To explain our approach, we consider two different discretizations of \(\Delta\) in the grid point \(\mathbf{x}_{0}\). If only the data at \(\mathbf{x}_{0},\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3}\) and \(\mathbf{x}_{4}\) are used, we employ the approximation  
\[\Delta u(\mathbf{x}_{0})\approx\sum_{j=0}^{4}a_{j}u(\mathbf{x}_{j}). \tag{2}\]  
Otherwise, if all the data in \(\mathbf{x}_{0},\mathbf{x}_{1},\ldots,\mathbf{x}_{8}\) are used, we change (2) accordingly to obtain  
\[\Delta u(\mathbf{x}_{0})\approx\sum_{j=0}^{8}a_{j}u(\mathbf{x}_{j}). \tag{3}\]  
Note that in the the presence of a uniform rectangular grid with the parameters \(h_{x}\) and \(h_{y}\), as shown in Figure 1, left, the optimal approximation in (2) can be given as  
\[\Delta u(\mathbf{x}_{0})\approx-(\frac{2}{h_{x}^{2}}+\frac{2}{h_{y}^{2}})u( \mathbf{x}_{0})+\frac{1}{h_{x}^{2}}u(\mathbf{x}_{1})+\frac{1}{h_{x}^{2}}u( \mathbf{x}_{3})+\frac{1}{h_{y}^{2}}u(\mathbf{x}_{2})+\frac{1}{h_{y}^{2}}u( \mathbf{x}_{4}), \tag{4}\]  
so that the error becomes \(\mathcal{O}(h_{x}^{2})+\mathcal{O}(h_{y}^{2})\), assuming that \(u\) is four times differentiable. By changing the geometry in Figure 1, left, in any way, this accuracy is lost. In a similar way, using a square grid with \(h=h_{x}=h_{y}\) and all the nine-point data in (3), the optimal approximation in (3) becomes  
\[ \begin{split}\Delta u(\mathbf{x}_{0})&\approx- \frac{20}{6h^{2}}u(\mathbf{x}_{0})+\frac{4}{6h^{2}}\cdot(u(\mathbf{x}_{1})+u( \mathbf{x}_{2})+u(\mathbf{x}_{3})+u(\mathbf{x}_{4}))\\ &+\frac{1}{6h^{2}}\cdot(u(\mathbf{x}_{5})+u(\mathbf{x}_{6})+u( \mathbf{x}_{7})+u(\mathbf{x}_{8})).\end{split}  \tag{5}\]  
In this case, provided that \(\Delta u=0\), the error becomes \(\mathcal{O}(h^{4})\). For further details and extensions, we refer to [20]. Again, for a general geometry, such as for the one shown on the right of Figure 1, this approximation order is no longer valid.