A difference finite element method based on nonconforming finite element methods for 3D elliptic problems

• Published in: Advances in computational mathematics Vol. 51; no. 1
• Main Authors: Song, Jianjian, Sheen, Dongwoo, Feng, Xinlong, He, Yinnian
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.02.2025
• Subjects: Discretization; Elliptic functions; Error analysis; Finite difference method; Finite element method; Gloss; Mathematics; Two dimensional analysis
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-025-10219-x

In this paper, a class of 3D elliptic equations is solved by using the combination of the finite difference method in one direction and nonconforming finite element methods in the other two directions. A finite-difference (FD) discretization based on $$P_1$$ P 1 -element in the z -direction and a finite-element (FE) discretization based on $$P_1^{NC}$$ P 1 NC -nonconforming element in the ( x ,  y )-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of $$P_1^{NC}$$ P 1 NC -nonconforming finite element methods in the 2D elliptic equation and the error estimation of the $${H^1}$$ H 1 -norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the $$P_1$$ P 1 element in the z -direction and the FE method based on the Crouzeix-Raviart element, the $$P_1$$ P 1 linear element, the Park-Sheen element, and the $$Q_1$$ Q 1 bilinear element, respectively, in the ( x ,  y )-plane.