A certified matrix defect transfers to a Galerkin error bound

For a standard near certificate, [F5] supplies the exact identity; a custom certificate supplies it as an audited hypothesis. In either case each exponent and transformed amplitude meet [F7], so the positive Duffy rule error is bounded by $E_m$. Add the separately certified nested fiber, node-evaluation and outward arithmetic radii with the triangle inequality; the certificate condition gives $\mathrm{\mid }{A}_{TS}-{\tilde{A}}_{TS}\mathrm{\mid }\le {\sum }_m\mathrm{\mid }{c}_m\mathrm{\mid }(E_m+{\rho }_m)={\epsilon }_{TS}^{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}\le {\tau }_{TS}$.

For a far certificate, [F6] bounds the difference between the exact entry and the positive centroid sum by ${\epsilon }_{TS}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}+{\epsilon }_{TS}^{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}}$. Under the strict ball condition, [F8] bounds the point-cluster error uniformly over the target ball. Multiplying by the positive target weights and adding the outward arithmetic radius gives the remaining terms in ${\epsilon }_{TS}^{\mathrm{f}\mathrm{a}\mathrm{r}}$. The total is at most ${\tau }_{TS}$ by the certificate condition; if $R\le a+b$, no FMM tail is used.

Apply [F1] with exact form $a$, exact load $F$, and conforming subspace $X_h$. It gives the unique exact solution $u$, the unique exact Galerkin solution $u_h$, and $\mathrm{\parallel }u-u_h{\mathrm{\parallel }}_H\le (1+C\mathrm{/}c){\inf }_{v_h\in X_h}\mathrm{\parallel }u-v_h{\mathrm{\parallel }}_H$, since the consistency defect is zero.

The perturbation bound and coercivity of $a$ imply ${\tilde{a}}_h(v_h,v_h)\ge (c-\delta )\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_H^2$ for all $v_h$. Also $\mathrm{\mid }{\tilde{a}}_h(v_h,w_h)\mathrm{\mid }\le (C+\delta )\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_H\mathrm{\parallel }{w}_h{\mathrm{\parallel }}_H$, and $\mathrm{\parallel }{\tilde{F}}_h\mathrm{\parallel }\le \mathrm{\parallel }{F}_h\mathrm{\parallel }+\eta$. Lax–Milgram [F2] therefore gives a unique perturbed solution ${\tilde{u}}_h$.

For a standard touching pair, [F5] gives finitely many smooth amplitudes with fixed derivative bounds. At level $k$, midpoint triangle subdivision and the $n=1$ pure-derivative remainder give total triangle error at most $C_{TS}{4}^{-k}$; where an opposite face has dimension three, its $2^k$ fiber midpoints add at most $C_{TS}{4}^{-k}$. The disjoint smooth fallback has the same bound after subdividing both reference triangles. For the far branch, [F6] gives centroid error at most $C_{TS}{2}^{-k}$. The fixed whole-panel balls give $q=a\mathrm{/}(R-b)<1$, and positivity fixes $Q=\mathrm{\mid }S\mathrm{\mid }\mathrm{/}(4\pi )$ and $W_T=\mathrm{\mid }T\mathrm{\mid }$ independently of level; hence [F8] gives a tail at most $C_{TS}{q}^{k+1}$ (zero if $a=0$). By [F4] the combined evaluation and arithmetic radius is at most $2^{-k-1}{\tau }_{TS}$, reached in finite precision refinement by hypothesis. Each total therefore tends to zero, so the first accepted level is finite for every ${\tau }_{TS}>0$. The ordered panel-pair set is finite, and a custom identity uses a supplied finite certificate rather than this loop. This proves the stated termination assertion.

Write $E_{TS}=A_{TS}-{\tilde{A}}_{TS}$. For coefficient vectors $c,d$, finite Cauchy–Schwarz gives $\mathrm{\mid }{d}^{T}Ec\mathrm{\mid }\le {\left({\sum }_{T,S}\mathrm{\mid }{E}_{TS}{\mathrm{\mid }}^2\right)}^{1\mathrm{/}2}\mathrm{\parallel }c{\mathrm{\parallel }}_2\mathrm{\parallel }d{\mathrm{\parallel }}_2$. The certified entry bounds imply $\mathrm{\mid }{E}_{TS}\mathrm{\mid }\le {\epsilon }_{TS}$. The supplied lower norm bound in [F4] gives $\mathrm{\parallel }c{\mathrm{\parallel }}_2\le {\gamma }_h^{-1}\mathrm{\parallel }{\sum }_S{c}_S{1}_S{\mathrm{\parallel }}_{H^{-1\mathrm{/}2}}$ and the corresponding bound for $d$. Substitute both inequalities and take the supremum over unit $H^{-1\mathrm{/}2}$ densities to obtain the bilinear operator-defect bound ${\delta }_h$. If a pair has no successful certificate, steps 1.1–1.2 provide no bound for that entry, so the procedure correctly returns uncertified.

Test the perturbed equation with ${\tilde{u}}_h$. Then $(c-\delta )\mathrm{\parallel }{\tilde{u}}_h{\mathrm{\parallel }}_H^2\le (\mathrm{\parallel }{F}_h{\mathrm{\parallel }}_{X_h^{\mathrm{\prime }}}+\eta )\mathrm{\parallel }{\tilde{u}}_h{\mathrm{\parallel }}_H$, so $\mathrm{\parallel }{\tilde{u}}_h{\mathrm{\parallel }}_H\le (\mathrm{\parallel }{F}_h{\mathrm{\parallel }}_{X_h^{\mathrm{\prime }}}+\eta )\mathrm{/}(c-\delta )$; this also holds if ${\tilde{u}}_h=0$.

Put $e_h=u_h-{\tilde{u}}_h$. Subtracting the perturbed equation from the exact discrete equation gives $a(e_h,w_h)=F_h(w_h)-{\tilde{F}}_h(w_h)+({\tilde{a}}_h-a)({\tilde{u}}_h,w_h)$. Take $w_h=e_h$. If $e_h\mathrm{\ne }0$, coercivity and the two defects give $\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_H\le c^{-1}(\eta +\delta \mathrm{\parallel }{\tilde{u}}_h{\mathrm{\parallel }}_H)$; if $e_h=0$ the same inequality holds. Substitute step 2.2 to obtain $\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_H\le c^{-1}(\eta +\delta (\mathrm{\parallel }{F}_h{\mathrm{\parallel }}_{X_h^{\mathrm{\prime }}}+\eta )\mathrm{/}(c-\delta ))$.

By the triangle inequality, $\mathrm{\parallel }u-{\tilde{u}}_h{\mathrm{\parallel }}_H\le \mathrm{\parallel }u-u_h{\mathrm{\parallel }}_H+\mathrm{\parallel }{e}_h{\mathrm{\parallel }}_H$. Apply step 1.3 and step 3.1 to get the displayed bound. In the BEM specialization, [F3] verifies the exact-form hypotheses and the entrywise and form estimates in steps 1.1–2.1 verify that the successful assembly defect is the $\delta$ required here.