Immersed finite element methods for convection diffusion equations

In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $L^2$ and $H^1$-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $H^1$ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.