# Glossary¶

## Equations¶

### Poisson Equation¶

Let the operator $$\Delta$$ be given by

$\Delta u = \frac{\partial^2 u}{\partial x_1^2} + \cdots + \frac{\partial^2 u}{\partial x_d^2} \,,$

where $$u$$ is a sufficiently smooth function. The Poisson equation is

$-\Delta u = f \,,$

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

$[R_\mathrm{inj} u](x) = u(x) \quad \text{for all} \quad x \in \mathcal{G}_{\mathbf{h}'} \,.$

### 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

$\begin{split}[P_\mathrm{inj} u](x) = \begin{cases} u(x) & \text{for } x \in \mathcal{G}_{\mathbf{h}'} \\ 0 & \text{otherwise} \end{cases} \quad \text{for all} \quad x \in \mathcal{G}_{\mathbf{h}} \,.\end{split}$

## Error Propagation Operators¶

### Coarse Grid Correction¶

The error propagator of the coarse grid correction is defined by

$E = I - P (I - E_c) L_c^{-1} R L \,,$

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.