Rayleigh-Ritz method

The Ritz method minimizes a quadratic energy over a finite-dimensional trial space. For basis functions ${\varphi }_j$ and bilinear form $a$, its linear system is $Ka=F$ with $K_{jk}=a({\varphi }_k,{\varphi }_j)$ and $F_j=⟨f,{\varphi }_j⟩$.

A trial function is a chosen candidate used to approximate a solution or evaluate a variational principle. For an eigenvalue upper bound from a Rayleigh quotient, it must be nonzero, have finite numerator and denominator, and lie in the appropriate variational domain. In a Dirichlet problem it satisfies the fixed endpoint values. Natural boundary conditions need not always be imposed on a trial in the quadratic-form domain, but a candidate violating an essential boundary condition does not give the stated bound.

For a self-adjoint Sturm-Liouville operator $L_{ϵ}=L_0+ϵL_1$ with common boundary conditions and an $L^2$-normalized lowest eigenfunction $U_0$ of $L_0$, evaluating the Rayleigh quotient on $U_0$ gives the displayed upper bound. More generally the lowest eigenvalue is the infimum of affine functions of $ϵ$, so it is concave in that parameter. The trial state must satisfy the boundary requirements of the variational domain; changing the operator without changing the trial function's normalization can still be legitimate when the weight is fixed.

For $-(p{y}^{\prime }{)}^{\prime }=\lambda wy$ on $[0,1]$ with $p=(1+x)e^{2x}$, $w=e^{2x}$, $y(0)=0$ and $y^{\prime }(1)=0$, the Rayleigh quotient is $\int p{y}^{\prime 2}/\int w{y}^2$. The trial $y_1=x{e}^{-x}$ gives numerator $5/12$ and denominator $1/3$, hence $0<{\lambda }_{\min }\le 5/4$. Every displayed $y_n$ obeys the same mixed boundary conditions; so does $\sin [(m+1/2)\pi x]$. Adding any of these to the trial space cannot worsen the optimized upper bound. Matrix elements $A_{ij}=\int p{y}_i^{\prime }{y}_j^{\prime }$ and $M_{ij}=\int w{y}_i{y}_j$ give a finite-dimensional generalized eigenvalue problem $Av=\lambda Mv$. The case $n=1$ repeats the original trial and adds no variational freedom.

For a bounded symmetric coercive bilinear form on a Hilbert space, a solution of $a(u,w)=\ell (w)$ minimizes $J(v)=a(v,v)/2-\ell (v)$. The displayed identity proves strict minimality and uniqueness. Restricting to a conforming finite element space makes the discrete Ritz method equivalent to the Galerkin method. Galerkin orthogonality gives exact best approximation in the energy norm; the Céa lemma translates it into a reference-norm estimate.

A continuous symmetric coercive bilinear form $a$ on a real Hilbert space and a continuous linear functional $\ell$ give a unique minimum of $I$. The parallelogram identity makes any minimizing sequence Cauchy in the energy norm; coercivity and completeness give its limit. Differentiation gives $a(u,w)=\ell (w)$, and $I(v)-I(u)=a(v-u,v-u)$ proves uniqueness. A nonsymmetric operator instead differentiates to its symmetric part. Mere pointwise positivity without a lower bound in the chosen norm need not yield a minimizing vector.

A symmetric coercive bilinear form $a$ defines the energy norm $\Vert v{\Vert }_a=\sqrt{a(v,v)}$. In this norm the Ritz method is an orthogonal projection onto its finite-dimensional trial space.