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

翻译：本研究深入探讨了通过有限元法求解二维泊松问题，该问题在热传导、静电学、重力势及流体动力学等多种物理场景中具有重要意义。然而，由于域内复杂性（如凹角、裂纹及边界处解的不连续性）以及奇异源函数的存在，寻找这些问题的精确解可能变得复杂且具有挑战性。本工作的重点在于求解在域顶点处存在凹角（部分内角大于180度）情况下的泊松方程。当域存在凹角时，数值解在角点附近可能呈现奇异行为。为解决此问题，我们提出了一种梯度网格算法，以处理奇异点附近的解。我们推导了H1和L2误差估计结果，并利用MATLAB展示了验证理论结果的数值解。通过探索这些概念，我们期望为泊松问题提供新的见解，并启发未来研究如何应用数值方法解决复杂物理场景。