求解非齐次跳跃界面问题的新型浸入有限元法
A New Immersed Finite Element Method for Interface Problems with Non-Homogeneous Jump Conditions
摘要: 针对非齐次跳跃条件的椭圆型界面问题，提出了新型浸入有限元方法。该方法基于非拟合网格，将分片线性多项式延拓到整个平面，用来逼近有限元函数。通过延拓点以及延拓点之间界面上的函数积分平均来求解分片线性多项式。数值算例验证了该方法的有效性。
Abstract: A new immersed finite element method is proposed to solve the elliptic interface problem with non-homogeneous jump conditions. This method is based on unfitted mesh, and the piecewise linear polynomial is extended to the whole plane to approximate the finite element function. The piecewise linear polynomial functions are solved by the extension points and the average of the integral of function on the interface between the extension points. Numerical examples verify the effectiveness of the method.
文章引用：张帅帅, 秦芳芳. 求解非齐次跳跃界面问题的新型浸入有限元法[J]. 理论数学, 2024, 14(6): 113-121. https://doi.org/10.12677/pm.2024.146232

参考文献

[1] Li, H., Jiang, G., Yang, K. and Gao, X. (2020) A New Approach to Solve Multi-Medium Nonlinear Transient Heat Conduction Problems Using Interface Integration BEM. Engineering Analysis with Boundary Elements, 119, 269-279. [Google Scholar] [CrossRef] 
[2] Zhang, T., Wu, J. and Lin, X. (2020) An Improved Diffuse Interface Method for Three-Dimensional Multiphase Flows with Complex Interface Deformation. International Journal for Numerical Methods in Fluids, 92, 976-991. [Google Scholar] [CrossRef] 
[3] Hu, X., Ni, G., Fan, Z., Gu, J. and Dai, Z. (2021) Algorithm of Radiation Hydrodynamics with Nonorthogonal Mesh for 3D Implosion Problem. Journal of Computational Physics, 437, Article 110309. [Google Scholar] [CrossRef] 
[4] Daniel Deborah, O. and Moyosola, A. (2020) Laplace Differential Transform Method for Solving Nonlinear Nonhomogeneous Partial Differential Equations. Turkish Journal of Analysis and Number Theory, 8, 91-96. [Google Scholar] [CrossRef] 
[5] 冯亚芳. 二阶椭圆界面问题浸入界面有限元方法的多重网格算法[D]: [硕士学位论文]. 南京: 南京师范大学, 2017.
[6] Chen, Z. and Zou, J. (1998) Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems. Numerische Mathematik, 79, 175-202. [Google Scholar] [CrossRef] 
[7] Sun, B. and Suo, Z. (1997) A Finite Element Method for Simulating Interface Motion—II. Large Shape Change Due to Surface Diffusion. Acta Materialia, 45, 4953-4962. [Google Scholar] [CrossRef] 
[8] Qin, F.F., Chen, J.R., Li, Z.L. and Cai, M.C. (2017) A Cartesian Grid Nonconforming Immersed Finite Element Method for Planar Elasticity Interface Problems. Computers & Mathematics with Applications, 73, 404-418. [Google Scholar] [CrossRef] 
[9] LeVeque, R.J. and Li, Z. (1994) The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources. SIAM Journal on Numerical Analysis, 31, 1019-1044. [Google Scholar] [CrossRef] 
[10] Guo, R., Lin, T. and Lin, Y. (2019) Approximation Capabilities of Immersed Finite Element Spaces for Elasticity Interface Problems. Numerical Methods for Partial Differential Equations, 35, 1243-1268. [Google Scholar] [CrossRef] 
[11] Heltai, L. and Rotundo, N. (2019) Error Estimates in Weighted Sobolev Norms for Finite Element Immersed Interface Methods. Computers & Mathematics with Applications, 78, 3586-3604. [Google Scholar] [CrossRef] 
[12] Lin, T., Lin, Y.P. and Zhang, X. (2015) Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems. SIAM Journal on Numerical Analysis, 53, 1121-1144. [Google Scholar] [CrossRef] 
[13] Wang, S.H., Wang, F. and Xu, X.J. (2020) A Robust Multigrid Method for One Dimensional Immersed Finite Element Method. Numerical Methods for Partial Differential Equations, 37, 2244-2260. [Google Scholar] [CrossRef] 
[14] Lin, T. and Zhuang, Q. (2020) Optimal Error Bounds for Partially Penalized Immersed Finite Element Methods for Parabolic Interface Problems. Journal of Computational and Applied Mathematics, 366, 112401. [Google Scholar] [CrossRef] 
[15] Guo, R.C., Lin, T. and Zhuang, Q. (2019) Improved Error Estimation for the Partially Penalized Immersed Finite Element Methods for Elliptic interface Problems. International Journal of Numerical Analysis and Modeling, 16, 575-589.
[16] Ji, H.F., Zhang, Q., Wang, Q.L. and Xie, Y.F. (2018) A Partially Penalised Immersed Finite Element Method for Elliptic Interface Problems with Non-Homogeneous Jump Conditions. East Asian Journal on Applied Mathematics, 8, 1-23. [Google Scholar] [CrossRef] 
[17] He, X.M., Lin, T. and Lin, Y.P. (2011) Immersed Finite Element Methods for Elliptic Interface Problems with Non-Homogeneous Jump Conditions. International Journal of Numerical Analysis and Modeling, 8, 284-301.
[18] Li, Z.L., Lin, T. and Wu, X.H. (2003) New Cartesian Grid Methods for Interface Problems Using the Finite Element Formulation. Numerische Mathematik, 96, 61-98. [Google Scholar] [CrossRef] 

为你推荐