Finite element method

The finite element method begins by partitioning the domain into a finite number of non-overlapping elements (e.g., triangles, quadrilaterals, or tetrahedra) that collectively form a mesh. ³ On each element, the solution is approximated using polynomial basis functions (shape functions) that are nonzero only on that element and its immediate neighbors. The original PDE is then transformed into a weak (integral) formulation, often via the Galerkin method of weighted residuals, which requires the residual to be orthogonal to the basis functions. ¹ This yields a system of linear (or nonlinear) algebraic equations where the unknowns are the coefficients of the basis functions at the nodes. The system is assembled by summing contributions from each element, and boundary conditions are applied before solving. The accuracy of the approximation depends on element size, polynomial order, and mesh quality.

The conceptual foundations of FEM trace to work by Richard Courant in 1943, who used piecewise linear functions on triangular subdomains to solve a torsion problem. ² In the 1950s, structural engineers at Boeing, led by M. J. Turner and R. W. Clough, independently developed the method for aircraft wing analysis; Clough coined the term “finite element” in 1960. ⁴ John H. Argyris and colleagues published a seminal series of papers on energy theorems and matrix methods, while Olgierd Zienkiewicz and Y. K. Cheung’s 1967 book “The Finite Element Method in Structural and Continuum Mechanics” established the technique as a general tool for PDEs. ² The method gained rapid acceptance after the development of digital computers, and by the 1970s it was applied to fluid dynamics, heat transfer, and nonlinear problems. ¹

FEM is the dominant method for structural analysis, including stress, vibration, and buckling of buildings, bridges, and aerospace components. In fluid dynamics, the finite element formulation is used for incompressible and compressible flows, often coupled with stabilization techniques. Heat transfer simulations rely on FEM to model conduction, convection, and radiation in complex geometries. ¹ Electromagnetics applications include antenna design, waveguide analysis, and eddy current modeling. ³ In biomechanics, FEM simulates bone stress, soft tissue deformation, and blood flow in arteries. The method is also used in geophysics, acoustics, and multiphysics coupling, where multiple physical phenomena interact (e.g., thermal–mechanical–electrical).

Before Courant, Alexander Hrennikoff in 1941 proposed a “framework method” using a network of bars to approximate plane elasticity, which is a precursor to FEM. ⁴ Variational crimes—such as using numerical integration or non-conforming elements—can introduce errors that are often overlooked in practice. ¹ The p-version of FEM increases polynomial order within an element rather than refining the mesh (h-version), and hp-adaptive methods combine both for exponential convergence. ⁵ Meshless methods (e.g., element-free Galerkin) avoid explicit element connectivity, offering advantages for crack propagation and large deformations. ⁵ A posteriori error estimation, such as the Zienkiewicz–Zhu error estimator, allows adaptive mesh refinement in commercial codes. FEM also underpins shape and topology optimization, where the material distribution within a design space is iteratively updated to meet performance targets. ³

Mesh

A discretization of the domain into a finite number of elements (e.g., triangles, quadrilaterals) that collectively cover the geometry and are used to define the approximation.

Basis function

A local polynomial function (shape function) associated with a node or element that approximates the solution within that element; typically nonzero only over a small region.

Weak formulation

An integral reformulation of a PDE obtained by multiplying by a test function and integrating by parts, which reduces the required smoothness of the solution.

Stiffness matrix

The assembled matrix of coefficients in the linear system arising from the finite element discretization of a PDE, relating nodal forces to displacements.

Element

A simple geometric cell (e.g., triangle, quadrilateral, tetrahedron) that forms a building block of the mesh and on which the approximation is defined.