The Immersed Discontinuous Galerkin Method for Elliptic Interface Problems
Provides a computationally efficient and theoretically rigorous IFE method for solving elliptic interface problems, which are important in materials science and fluid dynamics.
This paper constructs explicit immersed finite element (IFE) functions for elliptic interface problems that satisfy interface conditions exactly without solving auxiliary systems. The method achieves optimal convergence rates in H^1 and L^2 norms, with condition numbers robust to interface location.
This paper is devoted to construction and convergence analysis of the linear explicit immersed finite element (IFE) function. For the interface elements, the proposed IFE functions precisely satisfy the interface conditions on the actual interface. The IFE functions are constructed in an explicit form and can be obtained directly without solving any auxiliary problems or local linear systems. Although the constructed IFE functions are non-polynomial, we establish rigorous theoretical analysis showing that they achieve optimal approximation properties and satisfy the essential trace inequalities. And the constants in the analysis are independent of how the interface cuts through the elements. Based on these IFE functions, an immersed discontinuous Galerkin numerical scheme is developed. Several numerical experiments are implemented to confirm that both the IFE functions and the numerical method achieve optimal convergence rates in the $H^1$ and $L^2$ norms. Furthermore, the numerical results indicate that the condition numbers of the stiffness matrices are robust with respect to the interface location.