associated nodal basis functions. Then the interpolation operator is defined as $$I_{h}w(x):=\sum_{i=1}^{N_{h}}w(x_{i})\phi_{i}(x)  \ \  {\rm for \ any} \ w \in L^{2}(\Omega).$$

Moreover, about the interpolation error estimate, we have the following result.

\[lem:interpolation error estimate\][([@Ciarlet2002The], Thm. 3.1.6)]{} For all $w \in W^{k+1,p}(\Omega), k\geq 0, p, q\in[0,+\infty)$ and $0\leq m \leq k+1$, we have $$\|w-I_{h}w\|_{W^{m,q}(\Omega)}\leq c_{I}h^{k+1-m}\|w\|_{W^{k+1,p}(\Omega)},$$ where $c_{I}$ is a constant which is independent of the mesh size $h$.

In the classical finite element based algorithm, the discretization mesh size is fixed in advance. When computing on the finer mesh, the scale of the discretized problem becomes larger and the computation cost becomes larger. So it is important to consider reducing the computation cost. In the first several iterations of the algorithm, the iteration precision is not that good, which means using coarse mesh will not make the precision worse, but reduce the computation amount. While as the iteration process proceeds, the iteration precision becomes higher and higher. In this case, using finer mesh is necessary. Thus we introduce the idea of gradually refining the grid. Specifically, in the initial iteration we choose coarse mesh and obtain a solution first, then use the interpolation operator to project the obtained solution 