  In the context of unfitted finite element discretizations the realization of
high order methods is challenging due to the fact that the geometry
approximation has to be sufficiently accurate. Recently a new unfitted finite
element method was introduced which achieves a high order approximation of the
geometry for domains which are implicitly described by smooth level set
functions. This method is based on a parametric mapping which transforms a
piecewise planar interface (or surface) reconstruction to a high order
approximation. In the paper [C. Lehrenfeld, A. Reusken, \emph{Analysis of a
High Order Finite Element Method for Elliptic Interface Problems}, arXiv
1602.02970, Accepted for publication in IMA J. Numer. Anal.] an a priori error
analysis of the method applied to an interface problem is presented. The
analysis reveals optimal order discretization error bounds in the $H^1$-norm.
In this paper we extend this analysis and derive optimal $L^2$-error bounds.
