A numerical technique for approximating solutions of partial differential equations and variational problems by subdividing the domain into small, simple-shaped elements and using piecewise polynomial basis functions to build a global approximate solution.

Principle

Formulate the problem in a variational (weak) form on the domain, choose a finite-dimensional space spanned by local basis functions supported on mesh elements, assemble the resulting linear or nonlinear system by elementwise integrations, then solve for the coefficient vector that defines the approximate solution.

Demonstration

To approximate −Δu=f on a polygonal domain with Dirichlet boundary data, triangulate the domain, take piecewise-linear basis functions on the triangles, assemble the stiffness matrix by integrating gradients of basis pairs on each triangle and solve the resulting sparse linear system for nodal values approximating u.

Misapplication

Using overly coarse meshes without convergence checks produces misleading results; applying elementwise polynomial bases without ensuring inter-element continuity appropriate to the problem (e.g., using discontinuous bases where continuity is required) yields incorrect approximations.

Consequence

When mesh resolution, element order, and solver accuracy are chosen compatibly, the method converges to the true solution with predictable rates, yields sparse systems exploitable by efficient solvers, and adapts naturally to complex geometries.

Reversal

The opposite approach is global spectral approximation on the whole domain using globally supported basis functions; that can offer faster convergence for smooth coefficients but loses locality and mesh flexibility.

Boundary

Intended for boundary-value and variational problems on domains that admit convenient meshing; not a black-box for all PDEs — formulation must respect regularity, boundary conditions, and compatibility of function spaces.

Semantic Tension

Close alternatives include finite difference methods that discretize derivatives on grids (simple but less geometric flexibility) and spectral methods that use global bases (high accuracy for smooth problems but poor local resolution).

Synthesis

The finite element method converts a continuous variational problem into a finite-dimensional algebraic system by local approximation on mesh elements and assembly, balancing locality, geometric flexibility, and systematic convergence control.