Many engineering models are governed by differential equations on continuous bodies or regions. The finite element method (FEM) replaces that continuous problem with a finite collection of smaller pieces called elements connected at nodes.

This replacement is called discretization, and the resulting collection of elements and nodes is a mesh.

Elements approximate the field

Instead of solving for a displacement, temperature or other field at every point independently, FEM represents the field inside each element using a limited number of nodal values and interpolation rules.

Smaller or higher-order elements can usually represent more detailed variation, at greater computational cost.

Element types

Line elements are useful for idealized bars and beams, surface elements for thin plates and shells, and solid elements for three-dimensional bodies.

Choosing an element type also chooses modelling assumptions; a geometrically similar mesh can represent a different mathematical model if its element formulation is different.

Mesh quality and refinement

A mesh should resolve important geometry and field variation. Refining a mesh where gradients are strong can improve the approximation, but adding elements indiscriminately increases cost without guaranteeing a better model.