  Partial differential equations on manifolds have been widely studied and
plays a crucial role in many subjects. In our previous work, a class of
integral equations was introduced to approximate the Poisson problems on
manifolds with Dirichlet and Neumann type boundary conditions. In this paper,
we restrict our domain into a compact, two dimensional manifold(surface)
embedded in high dimensional Euclid space with Dirichlet boundary. Under such
special case, a class of more accurate nonlocal models are set up to
approximate the Poisson model. One innovation of our model is that, the normal
derivative on the boundary is regarded as a variable so that the second order
normal derivative can be explicitly expressed by such variable and the
curvature of the boundary. Our concentration is on the well-posedness analysis
of the weak formulation corresponding to the integral model and the study of
convergence to its PDE counterpart. The main result of our work is that, such
surface nonlocal model converges to the standard Poisson problem in a rate of
$\mathcal{O}(\delta^2)$ in $H^1$ norm, where $\delta$ is the parameter that
denotes the range of support for the kernel of the integral operators. Such
convergence rate is currently optimal among all integral models according to
the literature. Two numerical experiments are included to illustrate our
convergence analysis on the other side.
