Certified reduced order method for the parametrized Allen-Cahn equation

In this contribution, we propose and analyze an efficient reduced-order finite element method for the parametrized Allen-Cahn equation. First, for a given parameter, the equation is discretized by a stabilized semi-implicit scheme in time and finite element method in space. Then, the reduced basis is constructed by using proper orthogonal decomposition (POD) to an ensemble of snapshots, which are numerical solutions of the full discrete problem at some time instances and a number of parameters over the parametric domain. The main novelty of this work is the error analysis of the reduced-order model. For the first time an error estimate is derived for the reduced-order solution of the parametrized Allen-Cahn equation. To the best of our knowledge, no published paper takes into account the impact of the diffusion parameter when proving bounds on the errors. Moreover, we propose a POD technique using time difference quotients as snapshots to improve the error estimates with respect to time step. Finally, we provide several numerical examples to verify the theoretical error estimates.

Automated summary

In this paper, an efficient reduced-order finite element method is proposed and analyzed for the parametrized Allen-Cahn equation. The equation is first discretized using a stabilized semi-implicit scheme in time and a finite element method in space for a given parameter. Then, a reduced basis is constructed using proper orthogonal decomposition (POD) based on a set of snapshots. The main contribution of this work lies in the error analysis of the reduced-order model, where an error estimate is derived for the first time for the parametrized Allen-Cahn equation, considering the impact of the diffusion parameter.