Full Approximation Scheme (FAS)

Let the fine grid equations be written as

$$L^h ( u^h ) = f^h.$$ (1)

The FAS equations defined on the coarse grid are

$$L^H ( u^H ) = f^H$$ (2)

where,

$$f^H = I_h^H f^h + L^H (I_h^H \bar u^h ) - I_h^H L^h (\bar u^h),$$ (3)

and where $\bar u^h$ denotes an approximation to $u^h$, the exact solution of the fine grid problem. The coarse grid correction is done as

$$\bar u^h \leftarrow \bar u^h + I_H^h [ u^H - I_h^H \bar u^h ].$$ (4)

Note that $I_h^H u^h$ is a solution of the coarse grid equations. This means that once the solution of the fine grid problem was obtained, the coarse grid correction does not introduce any changes via interpolation. We regard this property as an essential one and in our derivation of the coarse grid optimization problem we make sure that it is satisfied.