Numerical investigation of convergence in the $L^{\mathrm{\infty }}$ norm for modified SGFEM applied to elliptic interface problems

Convergence in the $L^{\mathrm{\infty }}$ norm is a very important consideration in numerical simulations of interface problems. In this paper, a modified stable generalized finite element method (SGFEM) was proposed for solving the second-order elliptic interface problem in the two-dimensional bounded and convex domain. The proposed SGFEM uses a one-side enrichment function. There is no stability term in the weak form of the model problem, and it is a conforming finite element method. Moreover, it is applicable to any smooth interface, regardless of its concavity or shape. Several nontrivial examples illustrate the excellent properties of the proposed SGFEM, including its convergence in both the $L^2$ and $L^{\mathrm{\infty }}$ norms, as well as its stability and robustness.