Error analysis of a stable reduced order model based on the proper orthogonal decomposition method for the Allen–Cahn–Navier–Stokes system

Abstract

In this paper, we develop a stable reduced order model based on the proper orthogonal decomposition method, and conduct rigorous theoretical analyses for the Allen–Cahn–Navier–Stokes system. Firstly, the full order model and the reduced order model are constructed. The full order model uses finite element discretization for space and explicit–implicit discretization for time, where the same degree of Lagrange polynomials are used to discretize phase function ϕ, velocity u and pressure p for the spatial discretization. Such spatial discretization leads to unstable schemes due to the loss of inf–sup stability condition. Therefore, we consider adding some stability terms to increase the stability of the whole system. The local projection stabilization method is added to the gradient of velocity and pressure. It should be emphasized that the local projection stabilization method relaxes the constraints of the stabilization coefficient compared with the previous research. Secondly, on the basis of full order model, we obtain the reduced order model by using proper orthogonal decomposition method. Then, we theoretically validate the energy stabilities and the error estimate of the full order model and the reduced order model, respectively. After that, some numerical experiments are performed to verify the numerical accuracy and the computational efficiency of the reduced order model.