Journal
COMPUTERS & MATHEMATICS WITH APPLICATIONS
Volume 96, Issue -, Pages 67-76Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.camwa.2021.05.014
Keywords
Allen-Cahn model; POD technique; SFE; SROFE; Error estimates
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Funding
- National Science Foundation of China [11871122]
- Chongqing Education Board of Science Foundation [KJ1400602]
- Scientific and Technological Research Program of Chongqing Municipal Education Commission [KJQN201800818]
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This paper proposes a reduced-order finite element method based on the POD technique for simulating the Allen-Cahn phase field model, with error estimates of the solutions provided. Numerical results demonstrate the effectiveness of the proposed method.
In this paper, a reduced-order finite element (FE) method preserving the unconditional energy-stability is proposed to simulate the Allen-Cahn phase field model, based on the proper orthogonal decomposition (POD) method with the snapshot technique. We first derive the full order FE formulation of the Allen-Cahn model and compute its FE full solutions, from which we choose a few spatio-temporal solutions as snapshots. Based on the POD technique, we then build a set of optimal POD bases maximizing the energy content in the original ensemble data, and in the new low-dimensional space spanned by the POD bases, we establish a low-order numerical model of stable reduced-order FE (SROFE) formulation for the Allen-Cahn phase field model. We also prove error estimates of the SROFE solutions of the Allen-Cahn phase field model. Finally, some numerical results are provided to test the validity of the SROFE formulation.
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