  This paper considers the two-dimensional narrow escape problem in a domain
which is composed of a relatively big head and several thin necks. The narrow
escape problem is to compute the mean first passage time(MFPT) of a Brownian
particle traveling from inside the head to the end of the necks. The original
model for MFPT is to solve a mixed Dirichlet-Neumann boundary value problem for
the Poisson equation in the composite domain, and is computationally
challenging. In this paper, we compute the MFPT by solving an equivalent
Neumann-Robin type boundary value problem. By solving the new model, we obtain
the high order asymptotic expansion of the MFPT. We also conduct numerical
experiments to show the accuracy of the high order expansion. As far as we
know, this is the first result on high order asymptotic solution for NEP in a
general shaped domain with several absorbing neck windows. This work is
motivated by \cite{Li}, where the Neumann-Robin model was proposed to solve the
NEP in a domain with a single absorbing neck.
