FEMethode

FEMethode, or finite element method (FEM), is a numerical technique for solving boundary value problems that arise in engineering and physics. It discretizes a continuous domain into smaller elements and approximates the unknown field (such as displacement, temperature, or pressure) by simple shape functions defined on each element. By converting a differential equation into a variational (weak) form, the method yields a system of algebraic equations that can be assembled into a global stiffness (or conductance) matrix and a force vector. Solving this system provides approximate values at the mesh nodes, from which the field can be interpolated inside elements.

Typical steps include: constructing a mesh that subdivides the domain; choosing element types and interpolation orders (linear, quadratic, etc.); assembling the global matrices; applying boundary conditions and loads; solving the linear or nonlinear system; and post-processing results.

Historically, the method emerged in the 1940s–1960s for structural analysis and was developed in its modern form by researchers such as Courant, Kellogg, and later Zienkiewicz and others. It has since become a general-purpose tool for a wide range of physical phenomena, including solid mechanics, heat transfer, fluid dynamics, electromagnetics, and acoustics.

Advantages include the ability to handle complex geometries, heterogeneous materials, and nonlinear behavior; it is highly adaptable to irregular domains through meshing. Limitations involve the quality of the mesh, computational cost for large or nonlinear problems, and the need for careful validation and error estimation. Modern extensions include isogeometric analysis, adaptive mesh refinement, and discontinuous Galerkin formulations.

FEM is implemented in many commercial and open-source software packages such as ANSYS, ABAQUS, COMSOL, FEniCS, and FreeFEM, which provide pre-processing, solvers, and post-processing tools.