Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / B / 7 / c / Solution

Compute $C$ from part (b), solve the finite element equations, and calculate the element indicators ${\eta }_i^2=h_i^4\Vert R_h{\Vert }_{L^2(I_i)}^2$. Let $\eta =({\sum }_i{\eta }_i^2{)}^{1/2}$. Stop when $C\eta <\mathrm{TOL}$; the proved reliability bound then certifies the requested true-error tolerance.

Otherwise mark elements carrying a chosen fraction of the total indicator, bisect them, and repeat. To ensure termination as well as a valid stopping test, include a uniform-refinement safeguard, for example bisect every element on every second unsuccessful pass. This preserves adaptive refinement on the other passes while forcing $h_{\max }\to 0$.

Here termination can be justified directly. On each element $R_h=f+p^{\prime }{u}_h^{\prime }-q{u}_h$, so its global $L^2$ norm stays bounded if $\Vert u_h{\Vert }_{H^1}$ stays bounded. That stability follows from coercivity after subtracting the fixed lift one at the right endpoint. HenceThe safeguarded algorithm therefore eventually satisfies the strict tolerance. Reliable numerical evaluation of the residual integrals, or an upper bound for their quadrature errors, is part of using the certificate in an implementation.