Reduced basis method based on a posteriori error estimate for the parameterized Allen-Cahn equation

Abstract

In this paper we propose an efficient and accurate reduced-order method for the parameterized Allen-Cahn equation. The proposed method aims to construct efficient low-dimensional reduced basis to approximate the Allen-Cahn equation with desired accuracy for all parameters of interest. The key is to select minimal parameters for which the snapshots are collected to generate the reduced basis. The selection of these parameters is guided by a residual estimator. To this end, we first derive a posteriori error estimates for this residual estimator. Then, we propose a POD-greedy algorithm based on the derived a posteriori error estimates to construct the efficient reduced basis. The established a posteriori error estimate and the accuracy of the proposed reduced-order method are verified through several numerical examples. Specifically, our numerical experiments show that the obtained a posteriori error estimate is sharp and that the convergence of the error with respect to the POD-greedy iteration is exponential. In addition, the efficiency of the POD-greedy sampling procedure is demonstrated by some practical examples.