  In this paper, we present a nonlocal model for Poisson equation and
corresponding eigenproblem with Dirichlet boundary condition. In the direct
derivation of the nonlocal model, normal derivative is required which is not
known for Dirichlet boundary. To overcome this difficulty, we treat the normal
derivative as an auxiliary variable and derive corresponding nonlocal
approximation of the boundary condition. For this specifically designed
nonlocal mode, we can prove its well-posedness and convergence to the
counterpart continuous model. The nonlocal model is carefully designed such
that coercivity and symmetry are preserved. Based on these good properties, we
can prove the nonlocal model converges with first order rate in $H^1$ norm. Our
model can be naturally extended to Poisson problems with Robin boundary and
corresponding eigenvalue problem.
