Numerical analysis for computing multiple solutions of semilinear elliptic problems by high-index saddle dynamics: Part I--Index-1 case

This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the high-index saddle dynamics at the index-1 case, 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.