  A finite element method is introduced to track interface evolution governed
by the level set equation. The method solves for the level set indicator
function in a narrow band around the interface. An extension procedure, which
is essential for a narrow band level set method, is introduced based on a
finite element $L^2$- or $H^1$-projection combined with the ghost-penalty
method. This procedure is formulated as a linear variational problem in a
narrow band around the surface, making it computationally efficient and
suitable for rigorous error analysis. The extension method is combined with a
discontinuous Galerkin space discretization and a BDF time-stepping scheme. The
paper analyzes the stability and accuracy of the extension procedure and
evaluates the performance of the resulting narrow band finite element method
for the level set equation through numerical experiments.
