2.5.2.1.2. Finite Element Method

So, the most common discretization techniques available for the numerical solution of partial differential equations are the finite difference method, the finite element method, and the finite volume method. So, we will go into small details of this, so that you have a broader idea.

Discretization techniques help in converting continuous equations into a form that can be solved numerically. The finite difference method approximates derivatives using differences between grid points. The finite element method breaks down complex domains into simpler parts called elements, which can be solved individually. The finite volume method focuses on the conservation of fluxes through a control volume. Each of these methods has its applications depending on the problem's nature and the required accuracy.

Think of a complex landscape like a mountain range. Instead of trying to measure the entire landscape at once, we can measure small sections of the landscape one at a time. Each section is like an element in the finite element method, allowing us to understand the entire landscape by assembling the measurements from the smaller sections.

In the finite difference method, the flow field is dissected into a set of grid points. The continuous functions are approximated by discrete values of these functions calculated at grid points.

The finite difference method transforms differential equations into difference equations. By defining grid points in the flow field, we can calculate function values like velocity and pressure at these discrete points. For instance, if we have velocity defined at point 'i', we can estimate it at adjacent points 'i+1' and 'i-1', thus creating a network of values that can solve the problem iteratively.

Imagine you have a long road, and you want to measure the elevation along it. Instead of taking a continuous measurement, you decide to measure the elevation at specific points (or grid points) along the road. Each measurement represents a finite point on that road, helping you piece together the overall elevation profile.

In the finite element or finite volume method, the flow field is broken into smaller fluid elements called cells.

In the finite element method, the domain is divided into smaller, manageable parts called elements. Differential equations governing the behavior of each element are formulated and then solved numerically. This allows for greater accuracy and flexibility, especially in complex geometries. Each element’s behavior can be effectively described using simple equations, and the overall solution is obtained by combining the effects of all elements.

Consider building a complex puzzle. Each piece represents a finite element. By understanding how each piece fits with its neighbors, you can assemble the entire puzzle. This method allows you to work on simplified individual parts, simplifying a potentially overwhelming task into a series of smaller, more manageable challenges.

The process of discretization is known as grid generation or mesh generation.

Mesh generation involves creating a grid that represents the computational domain. This grid is crucial because the accuracy of the numerical solution depends on the fineness of the mesh. A finer mesh allows for better resolution of the flow characteristics but requires more computational resources. The goal is to balance accuracy and efficiency: a coarse mesh might miss important features, while a very fine mesh could slow down calculations significantly.

Think of a large piece of fabric you want to sew. If you want to create a detailed pattern, using a fine grid of points to guide where you place your stitches will ensure precision. However, if you only use a few points, your pattern may turn out blurry or imprecise, just like a coarse mesh might miss important flow features in computational fluid dynamics.

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