In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle $\omega>\frac{\pi}{2}$. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coefficients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In [11] we introduced a new partial differential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in [11] works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.