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.