Intro to Finite Elements

Approximates continuous fields with local functions over a mesh of elements.

Conceptual context

The finite-element method turns a continuous domain into a mesh of small regions called elements. Within each element, the physical field is represented by simple local functions. Shared nodes connect those local descriptions into a global solution satisfying equilibrium, continuity, and boundary conditions. The aim is not to copy the exact field point by point, but to build a controlled approximation space. As $h$ decreases, more degrees of freedom become available and $u_h$ can follow the physical solution more closely. Accuracy is assessed through $E_h$ and its rate of decrease, described by $p$.

🟢 Essential level

A finite element is a small part of the geometry where the field follows a simple approximation rule.

In a linear element, $N_1$ and $N_2$ blend $u_1$ and $u_2$ to construct $u_h$. Many connected pieces form a mesh; each contributes a local relation, and assembly links displacements to forces throughout the domain. Refinement means reducing $h$, which usually improves the representation of curves and gradients. Comparisons must retain the same loads, properties, and constraints so that the observed change measures discretization. Elements should also remain well shaped, with enough resolution wherever the field varies rapidly. The benefit still depends on solution smoothness, element quality, and boundary conditions. A denser mesh cannot repair an incorrect physical model: geometry, properties, and constraints must continue to represent the real system.

Choosing element type and mesh density is therefore a physical decision as much as a numerical one: a coarse mesh may be entirely adequate where the field is smooth, while a region with sharp gradients needs local refinement to avoid a misleadingly smooth-looking but inaccurate result.

🔵 Formal level

Basic formulas

Inside a linear element, the field is interpolated from nodal values and shape functions:

Linear interpolation in an element

The two nodal functions are defined over $\xi$ and form a partition of unity:

One-dimensional linear shape functions

For a linear axial bar, $E$, $A$, and $L_e$ determine the scale of $k_e$:

Element stiffness of a one-dimensional bar

Real student questions

Why use elements when differential equations already exist? Many geometries and boundary conditions lack manageable analytical solutions.

Are nodal values exact? They are coefficients of $u_h$ and require verification.

Do more elements always help? Only with a consistent formulation, adequate mesh quality, and controlled secondary errors.

Why can the matrix be singular? A rigid motion may remain free, or connectivity may be wrong.

What is the difference between h and p refinement? h-refinement reduces $h$; p-refinement raises polynomial order.

Which order should I observe? It depends on element type, norm, and solution smoothness.

Final synthesis

Finite elements approximate fields with local pieces coordinated globally. Accuracy improves as $h$ decreases, but convergence depends on regularity, mesh quality, formulation, and boundaries. Reliability comes from showing that $E_h$ decreases with a coherent $p$.