FEMs

Finite element methods (FEMs) are numerical techniques for solving boundary value problems for partial differential equations and integral equations. They achieve this by subdividing a complex domain into smaller, simple elements and by approximating unknown fields with interpolation functions defined on each element.

The typical workflow begins with discretizing the domain into elements (such as triangles and quadrilaterals in two dimensions, or tetrahedra and hexahedra in three dimensions), selecting shape functions, and formulating the weak (variational) form of the governing equations. Element contributions are assembled into a global system, usually written as K u = f, where K is the stiffness (and possibly mass and damping) matrix, u the nodal values, and f the applied loads and boundary conditions. After applying constraints, the resulting linear or nonlinear system is solved, and results are interpreted through post-processing to obtain quantities such as displacements, stresses, temperatures, or flow fields.

FEMs are widely used across engineering and physics to analyze structural mechanics, heat transfer, fluid dynamics, acoustics, electromagnetics, and multiphysics problems. They support steady-state and transient analyses, linear and nonlinear material behavior, and eigenvalue or modal analyses. Mesh quality and refinement strategies, including hp-adaptivity, strongly influence accuracy and computational cost.

The method was developed in the mid-20th century and has since become a fundamental tool in computational science, with a large ecosystem of software implementing its various formulations.