A modified finite volume element method for solving the phase field Allen-Cahn model

In this paper, a modified finite volume element (MFVE) method is studied for the nonlinear phase field Allen-Cahn model including a small perturbation parameter. To this end, a traditional finite volume element (FVE) algorithm is firstly proposed to the phase field Allen-Cahn equation and the error estimations of the FVE solutions are derived. After that, a set of optimal basis functions are obtained by using the proper orthogonal decomposition approach, and a MFVE method with the reduced order is established. In particular, the optimal error estimations of the MFVE solutions of the phase field equation are proved. Finally, the theoretical results are verified by numerical tests, which also show that the CPU running time of the MFVE algorithm is much smaller than that of the traditional FVE one. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates a modified finite volume element (MFVE) method for solving the nonlinear phase field Allen-Cahn model with a small perturbation parameter. A traditional finite volume element (FVE) algorithm is proposed for the phase field Allen-Cahn equation, and error estimations for the FVE solutions are derived. Optimal basis functions are obtained using the proper orthogonal decomposition approach, and a reduced order MFVE method is established. Theoretical results are verified through numerical tests, which also show the superior performance of the MFVE algorithm in terms of CPU running time.