FiniteElementmethode

FiniteElementMethode, commonly referred to as the Finite Element Method (FEM), is a numerical technique for solving boundary value problems for partial differential equations and integral equations. It approximates unknown fields by piecewise polynomial functions defined over a discretized domain called a mesh. The method relies on variational formulations to convert differential equations into a system of algebraic equations.

The domain is partitioned into finite elements (one-, two-, or three-dimensional) connected at nodes. Within each element, local shape functions interpolate the field, leading to the assembly of a global sparse stiffness and mass matrix and a load vector. Boundary conditions are imposed to yield a solvable linear or nonlinear system. Time-dependent problems use temporal discretization as needed.

Typical workflow: derive the weak form, select function spaces, generate a mesh, assemble element contributions into the global system, apply Dirichlet and Neumann boundaries, solve for nodal values, and post-process results. Accuracy improves with mesh refinement and higher-order elements; adaptive methods use error indicators to refine locally.

History and scope: FEM emerged in the mid-20th century from structural analysis and mathematical theory, with pivotal work by Clough, Zienkiewicz, Oden, and others. It has since become a standard tool across engineering, physics, and applied mathematics for linear and nonlinear, static and dynamic problems.

Applications and software: FEM is used in solid mechanics, heat conduction, fluid flow, electromagnetism, acoustics, and biomechanics. Implementations range from commercial packages (e.g., ANSYS, ABAQUS, COMSOL) to open-source libraries (FEniCS, deal.II, Elmer). Practitioners must consider mesh quality, element type, solver choice, and computational cost.