= Finite element method
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= FEM
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{synonym}
The <finite element method> replaces an infinite-dimensional <weak formulation> by finitely many unknowns attached to a <finite element mesh>. Simple local functions, often <piecewise polynomial functions>, join to form a global approximation. The attraction is that each <finite element> can be integrated separately, while the assembled equations enforce a global balance. A <piecewise-linear hat function> only meets its immediate neighbors, so a large problem can have a <sparse matrix> rather than a dense one. Gilbert Strang's MIT lecture develops this connection from the one-dimensional <weak formulation> through local hats to the <stiffness matrix> and load integrals.https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-fall-2008/676ba0a00eef45a6cb3823c4da61ee11_18-085F08-L18.pdf[Gilbert Strang, MIT 18.085, Lecture 18: Finite Elements in 1D, transcript pp. 2–8, 2008; accessed 2026-10-11]{ref}
For the conforming <Galerkin method>, let $V$ be a real <Hilbert space>, $a$ a bounded <coercive bilinear form>, and $\ell$ a bounded <linear functional>. Given a <conforming finite element space> $V_h\subset V$, the continuous and discrete problems are
$$
\begin{aligned}
u\in V:&\quad a(u,v)=\ell(v)&&\text{for every }v\in V,\\
u_h\in V_h:&\quad a(u_h,v_h)=\ell(v_h)&&\text{for every }v_h\in V_h.
\end{aligned}
$$
Writing $u_h=\sum_j c_j\phi_j$ in a <basis> of $V_h$ gives the <linear system>
$$
Kc=F,\qquad K_{ij}=a(\phi_j,\phi_i),\qquad F_i=\ell(\phi_i).
$$
Here $K$ is the <stiffness matrix> and $F$ is the vector of loads tested against the <basis> functions. The <Galerkin method> uses the same trial and test space in this construction. Finite element methods also use the <Petrov-Galerkin method>, with different trial and test spaces; testing against the trial space is therefore a property of this construction, rather than a definition of every <finite element method>.
The <finite element solution of the one-dimensional Poisson equation> below gives every entry explicitly. Refining the <finite element mesh> then improves the approximation by adding local degrees of freedom. For a symmetric <coercive bilinear form>, <Galerkin orthogonality> makes the discrete solution the best approximation in the <energy norm>; the geometry of <orthogonal projection> explains why this local construction yields a globally meaningful answer.
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