A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

doi:10.1007/s42967-023-00281-x

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (2): 992-1012.doi: 10.1007/s42967-023-00281-x

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A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

Baiying Dong^1,2, Zhilin Li³, Juan Ruiz-álvarez⁴

1. School of Civil and Hydraulic Engineering, NingXia University, Yinchuan 750021, Ningxia, China;
2. School of Mathematics and Computer Science, NingXia Normal University, Guyuan 756000, Ningxia, China;
3. Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA;
4. Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

Received:2022-10-30 Revised:2023-03-25 Accepted:2023-05-01 Online:2023-07-10 Published:2023-07-10
Contact: Zhilin Li,E-mail:zhilin@math.ncsu.edu E-mail:zhilin@math.ncsu.edu
Supported by:
B. Dong is partially supported by the National Natural Science Foundation of China (Grant No. 12261070) and the Ningxia Key Research and Development Project of China (Grant No. 2022BSB03048). Z. Li is partially supported by the Simons (Grant No. 633724) and by Fundación Séneca grant 21760/IV/22. J. Ruiz is partially supported by the Spanish national research project PID2019-108336GB-I00 and by Fundación Séneca grant 21728/EE/22. (Este trabajo es resultado de las estancias (21760/IV/22) y (21728/EE/22) financiadas por la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia con cargo al Programa Regional de Movilidad, Colaboración Internacional e Intercambio de Conocimiento “Jiménez de la Espada”. (Plan de Actuación 2022)).

Abstract

Abstract: In this paper, a new finite element and finite difference (FE-FD) method has been developed for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes. In the spatial discretization, the standard P₁ FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite, while near the interface, the maximum principle preserving immersed interface discretization is applied. In the time discretization, a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate. Correction terms are needed when the interface crosses grid lines. The moving interface is represented by the zero level set of a Lipschitz continuous function. Numerical experiments presented in this paper confirm second order convergence.

Key words: Anisotropic parabolic interface problem, Hybrid finite element and finite difference(FE-FD) discretization, Modified Crank-Nicolson scheme

Baiying Dong, Zhilin Li, Juan Ruiz-álvarez. A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 992-1012.

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A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

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