  This work delves into solving the two dimensional Poisson problem through the
Finite Element Method which is relevant in various physical scenarios including
heat conduction, electrostatics, gravity potential, and fluid dynamics.
However, finding exact solutions to these problems can be complicated and
challenging due to complexities in the domains such as re-entrant corners,
cracks, and discontinuities of the solution along the boundaries, and due to
the singular source function. Our focus in this work is to solve the Poisson
equation in the presence of re entrant corners at the vertices of domain where
some of the interior angles are greater than 180 degrees. When the domain
features a re entrant corner, the numerical solution can display singular
behavior near the corners. To address this, we propose a graded mesh algorithm
that helps us to tackle the solution near singular points. We derive H1 and L2
error estimate results, and we use MATLAB to present numerical results that
validate our theoretical findings. By exploring these concepts, we hope to
provide new insights into the Poisson problem and inspire future research into
the application of numerical methods to solve complex physical scenarios
