= Rayleigh-Ritz method
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{wiki=Rayleigh–Ritz_method}
= Ritz method
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{synonym}
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> $\ell$, take
$$
J(v)=\frac12a(v,v)-\ell(v).
$$
Writing $u_h=\sum_kc_k\phi_k$ in a trial <basis> and differentiating with respect to each coefficient gives
$$
Kc=F,\qquad K_{jk}=a(\phi_k,\phi_j),\qquad F_j=\ell(\phi_j).
$$
The same equations follow from testing the variational problem against each $\phi_j$, 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 $-u''=1$ on $(0,1)$ with zero <Dirichlet boundary conditions>, choose $a(v,w)=\int_0^1v'w'\,dx$, $\ell(v)=\int_0^1v\,dx$ and the one-dimensional trial space spanned by $\phi=x(1-x)$. 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.
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