FEM

FEM, or the finite element method, is a numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It subdivides a complex domain into smaller, simpler parts called elements, connected at nodes, and constructs approximate solutions from simple local functions defined on each element. The method reduces a continuous problem to a discrete system of equations, typically assembled into a global stiffness matrix and load vector.

In practice, the domain is meshed into elements such as triangles or quadrilaterals in 2D, tetrahedra or hexahedra in 3D. The solution within each element is approximated by shape functions (polynomials) that interpolate nodal values. Isoparametric elements use the same shape functions to map between reference and physical coordinates. Linear elements provide first-order accuracy; higher-order elements provide better accuracy at greater cost.

The typical workflow includes preprocessing (geometric modeling, material properties, boundary and initial conditions), meshing, solving the linear or nonlinear system Ku = f, and post-processing (displacements, stresses, heat flux). For time-dependent problems, FEM is combined with time-stepping schemes.

FEM is widely used in structural mechanics, heat transfer, fluid dynamics, electromagnetics, acoustics, and biomechanics. It supports static and dynamic analyses, eigenvalue problems, and coupled multiphysics simulations. Its advantages include suitability for complex geometries, material heterogeneity, and local refinement; limitations include mesh quality dependence, modeling errors, and high computational cost for large-scale nonlinear problems.

History: The method was developed independently in the 1940s and 1950s, with early contributions by Alexander Hrennikoff and Richard Courant, and was popularized for structural analysis by J. T. Clough in 1960.