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

TL;DR

The paper tackles the challenge of solving the 2D Poisson equation on non-convex polygonal domains, where re-entrant corners induce singularities that degrade standard finite element convergence. It introduces a graded-mesh strategy grounded in weighted (Kondrat’ev-type) Sobolev spaces to recover optimal $H^1$ and $L^2$ convergence, and provides explicit error estimates that quantify improved rates under corner singularities. Numerical results on convex and non-convex domains validate the theory, showing near-optimal rates for graded meshes and illustrating the limitations of uniform refinement in the presence of singularities. The study lays a foundation for extending graded-mesh techniques to 3D problems and higher-order elements, with potential applicability to a broad class of elliptic PDEs featuring singular behavior at domain corners.

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

Figures (10)

• Figure 1: Linear hat basis function in 2D
• Figure 2: Domain $\Omega$ containing one re-entrant corner.
• Figure 3: First row: the initial triangle and the midpoint refinement; second row: graded refinements ($\kappa_{Q_i}<0.5$).
• Figure 4: The new node on an edge $AB$. (left): $A \neq Q_i$ and $B \neq Q_i$ (midpoint); (right): $A=Q_i\; (\vert AB \vert = \kappa_{Q_i} \vert AB \vert ,\; \kappa_{Q_i} < 0.5$).
• Figure 5: (a) Initial mesh; (b) one refinement; (c) two refinements; (d) three refinements.

Theorems & Definitions (27)

• Remark 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 3.1
• Remark 3.2
• Lemma 3.3
• proof