Per definition, a mesh is a network that is formed of cells and points. It can have almost any shape in any size and is used to solve Partial Differential Equations. Each cell of the mesh represents an individual solution of the equation which, when combined for the whole network, results in a solution for the entire mesh.

Solving the entire object without dividing it into smaller pieces can be impossible because of the complexity that is within the object. Holes, corners and angles can make it extremely difficult for solvers to obtain a solution. Small cells on the other hand, are comparably easy to solve and therefore the preferred strategy.

## Brief History

The history of the mesh and meshing techniques is closely related to the history of numerical methods. The paper by Courant, Friedrichs and Lewy1

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is arguably the fundamental starting point for the Finite Differences Method (FDM), where concepts such as the C.F.L. (Courant, Friedrichs, Lewy) stability condition were introduced.

Historically, rectangular and Cartesian grids are associated with the Finite Differences, due to the easy identification of point neighbors, used in the derivatives approximations.

The Finite Element Method (FEM) allowed for mixed types of mesh cells, and unstructured meshes became feasible. Variational formulations being used to solve problems numerically can date back to Lord Rayleigh and Ritz works, from the end of the 19th century to the first decade of the 20th century2

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.

## Introduction

The first steps for numerically solving a set of partial differential equations (PDEs) is the discretization of the equations and the discretization of the problem domain. As mentioned earlier, solving the entire problem domain at once is impossible whereas solving multiple small pieces of the problem domain is perfectly fine. The equations discretization process is related to methods such as the Finite Differences Method, Finite Volume Method (FVM) and Finite Element Method, whose purpose is to take equations in the continuous form and generate a system of algebraic difference equations. The former, namely the domain discretization process, generates a set of discrete points and cells that cover the continuous problem domain.