A modified $P_1$ - immersed finite element method
This work provides a more accurate and efficient IFEM for solving elliptic interface problems, which is important for computational scientists and engineers dealing with discontinuous coefficients.
The authors propose an improved P1-based immersed finite element method (IFEM) for elliptic interface problems, adding line integral flux terms to achieve optimal H1 and L2 error estimates. Numerical experiments demonstrate robustness.
In recent years, the immersed finite element methods (IFEM) introduced in \cite{Li2003}, \cite{Li2004} to solve elliptic problems having an interface in the domain due to the discontinuity of coefficients are getting more attentions of researchers because of their simplicity and efficiency. Unlike the conventional finite element methods, the IFEM allows the interface cut through the interior of the element, yet after the basis functions are altered so that they satisfy the flux jump conditions, it seems to show a reasonable order of convergence. In this paper, we propose an improved version of the $P_1$ based IFEM by adding the line integral of flux terms on each element. This technique resembles the discontinuous Galerkin (DG) method, however, our method has much less degrees of freedom than the DG methods since we use the same number of unknowns as the conventional $P_1$ finite element method. We prove $H^1$ and $L^2$ error estimates which are optimal both in order and regularity. Numerical experiments were carried out for several examples, which show the robustness of our scheme.