 Rayleigh–Ritz method

The Rayleigh-Ritz method minimizes an energy functional over a finite-dimensional trial space. For a real Hilbert space $V$, a symmetric coercive bilinear form $a$ and a continuous linear functional $\ell$, takeWriting $u_h={\sum }_k{c}_k{\varphi }_k$ in a trial basis and differentiating with respect to each coefficient givesThe same equations follow from testing the variational problem against each ${\varphi }_j$, which is the Ritz-Galerkin equivalence for a symmetric coercive form. Nonsymmetric problems need a separate argument because differentiating $a(v,v)$ sees only its symmetric part.

For $-u^{\prime \prime }=1$ on $(0,1)$ with zero Dirichlet boundary conditions, choose $a(v,w)={\int }_0^1{v}^{\prime }{w}^{\prime }dx$, $\ell (v)={\int }_0^{1}vdx$ and the one-dimensional trial space spanned by $\varphi =x(1-x)$. Then $K=1/3$ and $F=1/6$, giving $u_h=x(1-x)/2$. Here the exact solution lies in the trial space, so minimization recovers it exactly; in general the Céa lemma controls the approximation error.