In this paper we provide a piecewise linear Galerkin approximation of a second order transmission problem across a highly conductive prefractal layer of von Koch type. We firstly generate an appropriate mesh adapted to the geometric shape of the interface and then we construct a refinement algorithm consistent with a suitable estimate in appropriate weighted Sobolev spaces. We also obtain a quasi-optimal error estimate in the energy norm and finally we demonstrate the validity of our theory through numerical tests. © 2012 IMACS.