The Immersed Skeletal Finite Element Method for Elliptic Interface Problems

In this paper, we propose a new immersed finite element (IFE) scheme that solves elliptic interface problems on unfitted meshes by combining the skeletal finite element method (FEM) with the standard FEM. We apply the skeletal FEM to interface elements, where both cell-wise and edge-wise piecewise polynomial functions are treated as unknowns. Specifically, we construct the cell-wise functions using IFE basis functions that exactly satisfy the interface conditions. The stabilizer can ensure the weak continuity between adjacent elements and is independent of the empirical parameters. For non-interface elements, we adopt the standard Lagrange FEM. Our scheme inherits the flexibility and parameter-free nature of the skeletal FEM and retains the simplicity and efficiency of the standard FEM. We rigorously prove that our method achieves optimal convergence rates in both the discrete H¹ norm and the L² norm, and we validate its accuracy and efficiency through numerical examples.