A Graded Mesh Refinement for 2D Poisson's Equation on Non Convex Polygonal Domains

TL;DR

This paper introduces a graded mesh refinement technique for solving the 2D Poisson equation on non-convex polygonal domains with re-entrant corners, improving accuracy near singularities.

Contribution

It proposes a novel graded mesh algorithm specifically designed to handle singularities at re-entrant corners in non-convex domains for the Poisson problem.

Findings

Derived H1 and L2 error estimates for the method.

Numerical results validate the theoretical error estimates.

The approach improves solution accuracy near singular points.

Abstract

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