Finite element method

Level SpecializationDifficulty ★★★★★Application⌖ Open in the map

What is it?

Solve a PDE on a complicated shape by splitting it into small elements, writing the weak form (integration by parts) and approximating the solution by piecewise polynomials. The standard tool of structural, thermal and electromagnetic engineering.

Formulas

-\Delta u = f \ \longrightarrow\ \int_\Omega\nabla u\cdot\nabla v\,\dd x = \int_\Omega f\,v\,\dd x\ \ \forall v \ \longrightarrow\ K\mathbf u = \mathbf f

The mathematics behind it

Integration by parts★★★★★fundamental
FEM rewrites $-u'' = f$ as $\int u'v' = \int fv$ by parts, lowering the smoothness required of $u$ .
Partial differential equations★★★★★fundamental
FEM is the dominant method to solve PDEs on complex geometries in engineering.
Numerical integration (quadrature)★★★★★frequent
FEM assembles element matrices by Gaussian quadrature on each element.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.