The FEM approach to the 2D Poisson equation in 'meshes' optimized with the Metropolis algorithm

TL;DR

This paper introduces a 2D mesh generation routine optimized with the Metropolis algorithm, enabling effective finite element analysis of the 2D Poisson equation on various domain shapes, including non-convex ones.

Contribution

It presents a novel mesh generation method optimized with the Metropolis algorithm for finite element analysis of the 2D Poisson problem.

Findings

Meshes can be generated with prescribed element size h.

The routine handles regular and non-regular domains.

Effective for solving the 2D Poisson equation on complex shapes.

Abstract

The presented article contains a 2D mesh generation routine optimized with the Metropolis algorithm. The procedure enables to produce meshes with a prescribed size h of elements. These finite element meshes can serve as standard discrete patterns for the Finite Element Method (FEM). Appropriate meshes together with the FEM approach constitute an effective tool to deal with differential problems. Thus, having them both one can solve the 2D Poisson problem. It can be done for different domains being either of a regular (circle, square) or of a non--regular type. The proposed routine is even capable to deal with non--convex shapes.