A local basis function and its associated geometric cell used in finite element discretizations; the term refers both to the cell (element) and the polynomial function space defined on it that together approximate fields over a mesh.

Principle

Partition of unity and locality: elements provide local supports for basis functions so global approximations are assembled from local contributions via variational formulations, enforcing continuity or conformity according to the chosen function space.

Demonstration

A linear Lagrange triangular finite element on a 2D mesh: each element carries three nodal basis functions that are linear on the triangle, enabling piecewise-linear approximation of a scalar field with inter-element continuity at nodes.

Misapplication

Mixing incompatible element function spaces (for example using continuous displacement elements with discontinuous pressure elements in incompressible elasticity) or increasing polynomial order without addressing mesh quality, leading to locking or non‑convergent approximations.

Consequence

Appropriate finite element choice yields controlled convergence rates, local error estimators, and stable coupling between fields (e.g., mixed formulations); it underpins solver design and informs mesh refinement and preconditioning strategies.

Reversal

Using global basis approaches (spectral methods with global trigonometric polynomials) reverses the local character of finite elements: high accuracy per degree of freedom may be obtained on smooth problems but locality and sparse assembly properties are lost.

Boundary

Refers to elements as combined geometric cell plus local function space within finite element methods; excludes purely finite difference stencils, meshless basis functions unless they are cast into element-like supports, and distinct discretizations like control volumes unless reinterpreted as elements.

Semantic Tension

Tension between 'finite element' as geometric cell and 'finite element' as basis function: practitioners may emphasize mesh geometry, polynomial order, or function-space conformity, creating different priorities in implementation and analysis.

Synthesis

A finite element is the localized unit of approximation combining a geometric cell and basis functions; assembling many such elements yields a global variational approximation whose accuracy and stability derive from local shape, order and inter-element continuity.