Title: 

Logarithmic Norms with Applications to Differential Operators

Abstract:

Originally defined for matrices in 1958, the logarithmic norm bounds solutions to initial value problems of the form $\dot{u} = Au + r$, $u(0) = u_0$, where $r$ is a forcing function or perturbation. Using differential inequalities one obtains
$$
\|u(t)\| \leq e^{tM[A]} \|u_0\| + \int_0^t e^{(t-s)M[A]} \|r(s)\| ds,
$$
where the logarithmic norm $M[A]$ is the best possible constant such that this bound holds for all $t \geq 0$. 

The logarithmic norm has since been extended to cover nonlinear maps and differential operators, unifying and quantifying notions such as definiteness for matrices; monotonicity for nonlinear maps; and ellipticity for differential operators. It plays an essential role in stability analysis, with broad and versatile applications in initial value problems, elliptic boundary value problems and initial-boundary value partial differential equations, as well as their discretizations.

Based on the new book [1], the talk gives a brief introduction to the essential background material, and proceeds to linear differential operators. The theory is exemplified by the 1D convection-diffusion operator, analyzing the solvability and conditioning of the associated equation. Strong ellipticity in $L^2$ may be lost for some combinations of boundary conditions and the Péclet number, causing rapidly increasing perturbation sensitivity and loss of stability. Similar issues are discussed for discretizations on nonuniform grids.

[1]  G. Söderlind, Logarithmic Norms, Springer Series in Computational Mathematics 63 (2024)