Finite element method

A numerical method for approximating solutions to boundary-value problems by dividing a domain into smaller elements and solving equations over them. It represents the solution locally and assembles those local relations into a global system.

Mesh generation: The process of partitioning a geometric domain into discrete elements connected at shared nodes. The mesh turns the continuous problem domain into the elements on which the method operates.
Partial differential equation: An equation involving an unknown function and its partial derivatives with respect to multiple variables. Many finite element problems begin as partial differential equations over a spatial or spacetime domain.
Structural analysis: The calculation of forces, stresses, and deformations in structures under loads. Finite elements let engineers predict how complex structures deform and carry load.
Finite difference method: A numerical method that approximates derivatives by differences between function values on a grid. Unlike finite elements, it typically approximates derivatives directly on a structured set of points.
A posteriori error estimation: The estimation of numerical error using a computed solution rather than only prior theoretical bounds. It identifies where a finite element mesh needs refinement after an initial solve.
Shape function: A basis function used to interpolate a field variable within an element from its nodal values. Shape functions define the local approximation assembled across the mesh.
Basis function: A function in a collection used to express or approximate other functions through linear combinations. Finite element approximations combine local basis functions to represent the unknown field.
Computational fluid dynamics: The numerical study of fluid flow using discretized equations of motion and transport. Finite element formulations solve fluid equations, especially in complex geometries and multiphysics settings.
Finite volume method: A numerical method that enforces conservation by balancing fluxes across the boundaries of discrete control volumes. Its conservation-centered control-volume balance differs from the finite element weak-form construction.
Adaptive mesh refinement: The selective modification of a computational mesh to improve accuracy or efficiency in regions of interest. Adaptive refinement addresses the cost of resolving sharp gradients with finite elements.

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