An enhanced flux continuity three-dimensional finite element method for heterogeneous and anisotropic diffusion problems on general meshes

• Published in: Journal of engineering mathematics Vol. 145; no. 1
• Main Authors: Hai, Ong Thanh, Nguyen, Thi Hoai Thuong, Le, Anh Ha, Do, Vuong Nguyen Van
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2024; Springer Nature B.V
• Subjects: Applications of Mathematics; Approximation; Computational Mathematics and Numerical Analysis; Continuity (mathematics); Finite element method; Linear functions; Mathematical analysis; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Theoretical and Applied Mechanics; General meshes; Heterogeneous anisotropic (possibly discontinuous) diffusion; 65N08; 76S05; 65N30; Finite element; Numerical flux continuity
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-024-10347-1

In this research, we present a novel enhanced flux continuity three-dimensional finite element method for heterogeneous and anisotropic (possibly discontinuous) diffusion problems on general meshes. We create a polygonal dual mesh T h ∗ and its submesh T h ∗ ∗ from a primal mesh T h in such a manner that a set number of adjacent tetrahedral elements of T h ∗ ∗ are united to form each dual control volume of T h ∗ , which corresponds to a primal vertex. The weak solution of the diffusion problem is approximated by the piecewise linear functions on the subdual mesh T h ∗ ∗ . In order to capture the local continuity of numerical fluxes across the interfaces, the proposed scheme gives the auxiliary face unknowns interpolated by the multi-point flux approximation. Moreover, the consistency, coercive, and convergence properties of the method are presented within a rigorous theoretical framework. Numerical results are carried out to highlight accuracy and efficiency.