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

A locally modified second-order finite element method for interface problems and its implementation in 2 dimensions