An Immersed Raviart–Thomas Mixed Finite Element Method for Elliptic Interface Problems on Unfitted Meshes

• Published in: Journal of scientific computing Vol. 91; no. 2; p. 66
• Main Author: Ji, Haifeng
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2022; Springer Nature B.V
• Subjects: Algorithms; Approximation; Basis functions; Computational Mathematics and Numerical Analysis; Fines & penalties; Finite element analysis; Finite element method; Interfaces; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Theoretical; 65N15; Interface problem; Mixed finite element; Immersed finite element; Unfitted mesh; 35R05; 65N30
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-022-01839-2

This paper presents a lowest-order immersed Raviart–Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element (IFE) is constructed by modifying the traditional Raviart–Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal finite element error estimates are proved rigorously with the constant independent of the interface location relative to the mesh. Some numerical examples are provided to validate the theoretical analysis.