Numerical analysis for saddle dynamics of some semilinear elliptic problems

TL;DR

This paper develops and analyzes numerical methods for computing saddle points of semilinear elliptic PDEs using saddle dynamics, ensuring accuracy, stability, and index-preservation of the schemes.

Contribution

It introduces a continuous-in-space saddle dynamics formulation for semilinear elliptic problems and provides rigorous analysis of finite element schemes for these systems.

Findings

Finite element schemes are stable and accurate.

Error estimates confirm index-preservation.

The methods effectively handle coupling and nonlocality.

Abstract

This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the index-1 saddle dynamics, or equivalently, the gentlest ascent dynamics. To establish clear connections between saddle dynamics and numerical methods of PDEs, as well as improving their compatibility, we first propose the continuous-in-space formulation of saddle dynamics for semilinear elliptic problems. This formulation yields a parabolic system that converges to saddle points. We then analyze the well-posedness, $H^1$ stability and error estimates of semi- and fully-discrete finite element schemes. Significant efforts are devoted to addressing the coupling, gradient nonlinearity, nonlocality of the proposed parabolic system, and the impacts of retraction due to the norm constraint. The error estimate results demonstrate the accuracy and index-preservation of the discrete schemes.