The Multigrid method based on addictive corrections bases on the original linear system to set up correction equations for the variables of the problem. These equations have as main objective to obtain new values for the variables, but that continue obeying the conservation equations, for more information see section 3. 

Another characteristic of these correction equations is that they are not just applied for one control volume, but for a series of control volumes. By that the linear system that should be solved to obtain the corrections is smaller than the one of the original linear system. 

The size difference between the original linear system and the resolved systems for the corrections calculation is that gives the term Multigrid, because the process described above can be understood as if the solver was solving the same problem in different meshes sizes. 

The illustration below illustrates what could be interpreted as being the different meshes created by the solver to accelerate the convergence process of the linear system.

The mesh of the Fig. 10.1.1.1 (a) represents the original problem, that creates a linear system in the form:

(10.1.1.1)

where [X] it is the vector that stores the 64 incognita of the mesh. 

After the solver determines the control volumes that will share the same correction equation. This grouping for correction equations creates the mesh 4x4 of the Figure 1 (b). 

It can be noticed, therefore, that the control volumes number 1, 2, 9 and 10 of the initial mesh will be subject to the same correction, which is identified with the number 1 in the mesh 4x4 of the Figure 1 (b). 

The operation principle of the Multigrid solver can be applied again on the linear system of the correction equations, represented by the mesh of the Figure 1(b), creating the mesh 2x2 of the Figure 1 (c), and so forth.

To accelerate the convergence process, the Multigrid solver approximates the value of the variables to the searched value concentrating on the resolution of the linear systems of the correction equations (rude meshes), instead of lose an enormous computational effort devoted to the resolution of the original linear system (refined meshes). 

The sequence adopted by the solver in the resolution of these linear systems of different sizes characterizes the different Multigrid cycles, commented in the next section.