Immersed finite element methods for linear and quasi-linear elliptic interface problems

We present a framework for constructing immersed finite element (IFE) spaces for second-order elliptic interface problems on interface-independent meshes exhibiting optimal convergence rates of $O\left(h^{p+1}\right)$ in the $L^2$ norm and $O\left(h^p\right)$ in the $H^1$ norm using a polynomial degree p. We consider linear problems with variable coefficients as well as quasi-linear problems. We present algorithms to create IFE spaces on elements cut by the interface and algorithms to solve the resulting finite element problems. We also present numerical results to demonstrate the performance of the IFE method for several linear problems with variable coefficients and quasi-linear elliptic interface problems in divergence form.