(Lax-Milgram and Ritz theory) Assume the followings: . is a bounded smooth domain in R³. X := := H↓(N) with the norm ||u|| = √√√Vu|² + |u|²dx. • {Xn} is a sequence of finite dimensional subspace of X such that Xn C Xn+1 and U1Xn = X. {j= 1,..., Nn} is a basis of Xn. a(,): X X X → R is symmetric, bilinear and 3||u||² ≤ a(u, u) & a(u, v) ≤ 10||u||||v|| Vu, ve X b(): X → R is linear functional such that |b(u)| ≤ ||u||. Þ(u) = a(u, u) - b(u) for u € X. := Assume un € Xn satisfies Þ(un) = infvɛx„ Þ(v) and u € X satisfies Þ(u) = infvɛx Þ(v). Prove that 1 5 || 2 10 - Un||² ≤ Þ(Un) — Þ(u)
