finiteElemente

Finite elements, or the finite element method (FEM), is a numerical approach for solving boundary value problems described by partial differential equations. The method works by partitioning a complex domain into smaller, simple pieces called elements, connected through nodes. Within each element, the unknown field is approximated by simple polynomial shape functions, and the local solutions are assembled to form an approximate global solution.

FEM relies on a variational (weak) formulation of the governing equations. The PDE is converted into a system of algebraic equations by integrating over the elements and incorporating material properties and boundary conditions. A mesh generator creates the discretization, selecting element shapes (triangles, quadrilaterals, tetrahedra, hexahedra) and orders (linear, quadratic, higher-order).

Local element equations are assembled into a global stiffness (or system) matrix. Boundary conditions are applied, and the resulting linear or nonlinear system is solved for the nodal values of the unknown field. Post-processing yields derived quantities such as stresses, fluxes, or temperatures.

Mesh quality and element type influence accuracy. Techniques such as h-refinement, p-refinement, and hp-adaptivity are used to improve precision. Isoparametric elements allow nonlinear element geometry to be represented with the same shape functions as the solution.

FEM is widely used in engineering and physics, including structural mechanics, heat conduction, fluid dynamics, electromagnetics, and acoustics. The method originated in the 1950s–1960s, with early work by Clough and later contributions by Zienkiewicz and others; it became widely adopted in the 1970s onward.