# Glossary¶

## Equations¶

### Poisson Equation¶

Let the operator \(\Delta\) be given by

where \(u\) is a sufficiently smooth function.
The *Poisson equation* is

where \(f\) is a function.

## Injection Operators¶

### Injection Restriction¶

Assume we have a fine grid \(\mathcal{G}_\mathbf{h}\) and a coarse grid
\(\mathcal{G}_{\mathbf{h}'}\) which is a subset of the fine grid.
The *injection restriction operator*
\(R_\mathrm{inj}: \ell_2(\mathcal{G}_\mathbf{h}) \to \ell_2(\mathcal{G}_{\mathbf{h}'})\)
is defined by

### Injection Interpolation¶

Assume we have a fine grid \(\mathcal{G}_\mathbf{h}\) and a coarse grid
\(\mathcal{G}_{\mathbf{h}'}\) which is a subset of the fine grid.
The *injection interpolation operator*
\(P_\mathrm{inj}: \ell_2(\mathcal{G}_{\mathbf{h}'}) \to \ell_2(\mathcal{G}_\mathbf{h})\)
is defined by

## Error Propagation Operators¶

### Coarse Grid Correction¶

The error propagator of the *coarse grid correction* is defined by

where

- \(L\) is the linear system operator on the fine grid,
- \(L_c\) is the linear system operator on the coarse grid,
- \(P\) is the interpolation operator,
- \(R\) is the restriction operator
- \(E_c\) is the error propagator of the method that solves the coarse grid equation. In case of a two-grid method, \(E_c = 0\).

## Software¶

### Matplotlib¶

Matplotlib is a Python library for visualizing mathematical functions and data.

- Homepage: https://matplotlib.org/