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/