  In this paper, we define new unfitted finite element methods for numerically
approximating the solution of surface partial differential equations using bulk
finite elements. The key idea is that the $n$-dimensional hypersurface, $\Gamma
\subset \mathbb{R}^{n+1}$, is embedded in a polyhedral domain in $\mathbb
R^{n+1}$ consisting of a union, $\mathcal{T}_h$, of $(n+1)$-simplices. The
finite element approximating space is based on continuous piece-wise linear
finite element functions on $\mathcal{T}_h$. Our first method is a sharp
interface method, \emph{SIF}, which uses the bulk finite element space in an
approximating weak formulation obtained from integration on a polygonal
approximation, $\Gamma_{h}$, of $\Gamma$. The full gradient is used rather than
the projected tangential gradient and it is this which distinguishes \emph{SIF}
from the method of [42]. The second method, \emph{NBM}, is a narrow band method
in which the region of integration is a narrow band of width $O(h)$. \emph{NBM}
is similar to the method of [13]. but again the full gradient is used in the
discrete weak formulation. The a priori error analysis in this paper shows that
the methods are of optimal order in the surface $L^{2}$ and $H^{1}$ norms and
have the advantage that the normal derivative of the discrete solution is small
and converges to zero. Our third method combines bulk finite elements, discrete
sharp interfaces and narrow bands in order to give an unfitted finite element
method for parabolic equations on evolving surfaces. We show that our method is
conservative so that it preserves mass in the case of an advection diffusion
conservation law. Numerical results are given which illustrate the rates of
convergence.
