Céa's lemma

Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. Let V {displaystyle V} be a real Hilbert space with the norm ‖ ⋅ ‖ . {displaystyle |cdot |.} Let a : V × V → R {displaystyle a:V imes V o mathbb {R} } be a bilinear form with the properties Let L : V → R {displaystyle L:V o mathbb {R} } be a bounded linear operator. Consider the problem of finding an element u {displaystyle u} in V {displaystyle V} such that Consider the same problem on a finite-dimensional subspace V h {displaystyle V_{h}} of V , {displaystyle V,} so, u h {displaystyle u_{h}} in V h {displaystyle V_{h}} satisfies By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that That is to say, the subspace solution u h {displaystyle u_{h}} is 'the best' approximation of u {displaystyle u} in V h , {displaystyle V_{h},} up to the constant γ / α . {displaystyle gamma /alpha .}