A reduced-order finite element method based on proper orthogonal decomposition for the Allen-Cahn model

Abstract

We focus on a reduced-order finite element (FE) method for the Allen-Cahn equation. Toward this end, we first derive a traditional FE (TFE) formulation for the Allen-Cahn equation and the energy stability and error estimates of the TFE solutions. Then we use the proper orthogonal decomposition (POD) technique to generate a set of POD bases from a few TFE solutions, establish a novel reduced-order FE (ROFE) formulation for the Allen-Cahn equation by the set of POD bases, and discuss the energy stability and error estimates of the ROFE solutions. Finally, we provide some numerical experiments to confirm the validity of the novel ROFE formulation.