Rigorous a-posteriori analysis using numerical eigenvalue bounds in a surface growth model

TL;DR

This paper develops a rigorous a-posteriori method using numerical eigenvalue bounds to prove global existence and uniqueness of smooth solutions for a nonlinear fourth-order PDE, ensuring the bounds are computationally feasible.

Contribution

It introduces a new approach combining numerical eigenvalue bounds with a-posteriori analysis to establish global solution properties of complex PDEs.

Findings

Derived rigorous upper bounds on the linearized operator's numerical range

Applied bounds in each time-step of discretization for global solution analysis

Established conditions for global existence and uniqueness of solutions

Abstract

In order to prove numerically the global existence and uniqueness of smooth solutions of a fourth order, nonlinear PDE, we derive rigorous a-posteriori upper bounds on the supremum of the numerical range of the linearized operator. These bounds also have to be easily computable in order to be applicable to our rigorous a-posteriori methods, as we use them in each time-step of the numerical discretization. The final goal is to establish global bounds on smooth local solutions, which then establish global uniqueness.