The Finite Element Method
Finite elements represent the solution as a sum of local basis functions and enforce the equation in a weighted-average (weak) sense.
A basis-function view of PDEs
The finite element method (FEM) partitions the domain into elements (triangles, quadrilaterals, tetrahedra, hexahedra) and approximates the solution as a linear combination of basis functions, each supported on a small patch of elements. Instead of enforcing the differential equation pointwise, FEM enforces it in a weak form: multiply by a test function, integrate over the domain, and integrate by parts to move derivatives onto the test function and reduce continuity requirements on the solution.
Substituting the basis expansion and choosing test functions from the same space (the Galerkin choice) yields a linear system K u = f, where K is the stiffness matrix assembled element by element. The result is a sparse system whose structure mirrors mesh connectivity.
Why weak forms help
The weak formulation lowers the order of derivatives that must exist, so solutions with kinks or material interfaces are admissible. It also gives clean, natural treatment of Neumann (flux) boundary conditions, which appear as boundary integrals rather than as extra constraints. Dirichlet conditions are imposed on the trial space directly.
Order, elements, and error control
Accuracy is raised by refining the mesh (h-refinement) or increasing polynomial degree (p-refinement); combining both (hp-refinement) can give exponential convergence for smooth solutions. Rigorous a posteriori error estimators drive adaptive refinement toward regions where the error is largest.
- Basis functions: Lagrange (nodal), hierarchical, and spectral variants
- Element types spanning simplices and tensor-product shapes
- Isoparametric mapping to represent curved boundaries accurately
- Mass and stiffness matrices assembled from local element contributions
FEM is the workhorse for structural mechanics, electromagnetics, and magnetohydrodynamic equilibrium and stability. In fusion device modeling it handles the shaped, curved geometries of vessels, coils, and blankets naturally, and its variational structure supports rigorous error estimates for design-relevant quantities.