  The locally modified finite element method, which is introduced in [Frei,
Richter: SINUM 52(2014), p. 2315-2334], is a simple fitted finite element
method that is able to resolve weak discontinuities in interface problems. The
method is based on a fixed structured coarse mesh, which is then refined into
sub-elements to resolve an interior interface.
  In this work, we extend the locally modified finite element method {in two
space dimensions} to second order using an isoparametric approach in the
interface elements. Thereby we need to take care that the resulting curved
edges do not lead to degenerate sub-elements. We prove optimal a priori error
estimates in the $L^2$-norm and in a discrete energy norm. Finally, we present
numerical examples to substantiate the theoretical findings.
