Certified near-quadrature and far-multipole matrix assembly

Assume the Axiom of Choice and Countable Choice. Let ${\mathcal{T}}_h$ and $X_h$ be the surface mesh and P0 space from Shape-regular surface meshes and conforming P0 densities. For an ordered target/source pair $(T,S)$ define its exact 3D Laplace single-layer matrix entry by $A_{TS}=⟨1_T,V{1}_S{⟩}_{\mathrm{\Gamma }}={\int }_T{\int }_S\frac{1}{4\pi \mathrm{\mid }x-y\mathrm{\mid }}d{S}_{y}d{S}_x,$ where the equality is proved for every panel pair by Certified touching-panel and disjoint-panel quadrature for the Laplace P0 matrix. Supply a positive per-entry tolerance ${\tau }_{TS}$ and construct exactly one auditable certificate as follows.

Process the finite set of ordered panel pairs once in lexicographic mesh order. For each pair, first certify whether the panels coincide, share an edge, share a vertex, or are disjoint; if the incidence is unresolved, return uncertified. Use exact algebraic panel coordinates or provide the needed certified incidence, separation (for disjoint panels), denominator, and derivative bounds, together with outward arithmetic whose radii can be refined to zero. A missing required bound returns uncertified. After selecting a standard branch below, refine its quadrature and arithmetic certificates by levels $k=0,1,2,\dots$ and accept the first level whose total error bound is at most ${\tau }_{TS}$. At each level refine precision until the combined node-value, evaluation and outward-arithmetic contribution to the entry bound is at most $2^{-k-1}{\tau }_{TS}$; charge the fiber-quadrature remainder separately.

• Near certificate. For coincident panels or panels sharing an edge or vertex, use the finite sign-cell/pulling-simplex identity of Certified touching-panel and disjoint-panel quadrature for the Laplace P0 matrix. A disjoint pair may also use that lemma's smooth fallback whenever the whole-panel far-cluster condition below is unavailable. Each term is a reference-triangle integral ${\int }_R\mathrm{\mid }z{\mathrm{\mid }}^{-{\alpha }_m}{q}_m(z)dz$ with ${\alpha }_m<2$ and transformed amplitude $G_m\in C^{2n}([0,1{]}^2)$. Record the exact change-of-variables derivation, the supplied derivative bounds, and the positive Gaussian remainder $E_m$ from The Duffy weighted product rule has an explicit bound using pure coordinate derivatives. If ${\rho }_m$ is the separately certified total of nested fiber, node-value and outward arithmetic radii, require ${\epsilon }_{TS}^{\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}}={\sum }_m\mathrm{\mid }{c}_m\mathrm{\mid }(E_m+{\rho }_m)\le {\tau }_{TS}$. A custom near identity may be accepted as a supplied finite certificate only after its identity, derivative bounds, every Gaussian hypothesis, and total error at most ${\tau }_{TS}$ have been verified; otherwise return uncertified. Do not run the standard refinement loop on a custom identity. For the standard panel-pair construction, every final exponent is ${\alpha }_m=0$.

• Far certificate. For disjoint panels, fix certified balls containing the entire source and target panels. Choose positive centroid rules on finite midpoint subdivisions and use A positive far-panel quadrature residual and normalized cluster combination to record the physical-kernel quadrature residual ${\epsilon }_{TS}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}$ and any separate outward evaluation radius ${\epsilon }_{TS}^{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}}$. If the source nodes with physical charges $q_j=v_j\mathrm{/}(4\pi )$ lie in $B(c,a)$ and the target nodes lie in $B(d,b)$ with the strict condition $R=\mathrm{\mid }d-c\mathrm{\mid }>a+b$, evaluate the single-cluster field of Single-cluster three-dimensional Laplace multipole evaluator. For $Q={\sum }_j\mathrm{\mid }{q}_j\mathrm{\mid }$ and target weight sum $W_T$, record ${\epsilon }_{TS}^{\mathrm{f}\mathrm{a}\mathrm{r}}={\epsilon }_{TS}^{\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{d}}+{\epsilon }_{TS}^{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}}+W_T\frac{Q}{R-b}\frac{(a\mathrm{/}(R-b){)}^{p+1}}{1-a\mathrm{/}(R-b)}+{\epsilon }_{TS}^{\mathrm{F}\mathrm{M}\mathrm{M}-\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}}\mathrm{.}$ The final term is the outward arithmetic radius of the finite cluster evaluation; it is zero only for an exact-arithmetic evaluation. At level $k$, use $m_T=m_S=p=k$ and refine outward precision as needed. If the whole-panel balls fail $R>a+b$, do not use this tail and instead use the disjoint smooth near fallback with $n=1$, subdividing both reference triangles at level $k$. No global multilevel or $O(N)$ work claim is made.

For a touching pair, use the standard identity and take $n=1$ in the weighted product rule on each similar-triangle subdivision at level $k$; when the opposite face has dimension three use $2^k$ midpoint subintervals for the finite fiber integral. Use midpoint subdivision, so each reference-triangle diameter is $2^{-k}$ times its original diameter, and use the per-level precision budget above. If a custom identity or any required hypothesis cannot be verified, return uncertified.

For every pair require its certified total error ${\epsilon }_{TS}\le {\tau }_{TS}$. Given one successful certificate for every ordered pair, form ${\tilde{A}}_h=({\tilde{A}}_{TS})$. If ${\gamma }_h>0$ is a certified lower norm bound satisfying $\mathrm{\parallel }c{\mathrm{\parallel }}_2\le {\gamma }_h^{-1}{\parallel {\sum }_T{c}_T{1}_T\parallel }_{H^{-1\mathrm{/}2}(\mathrm{\Gamma })},$ compute the returned defect certificate ${\delta }_h={\gamma }_h^{-2}{\left({\sum }_{T,S}{\epsilon }_{TS}^2\right)}^{1\mathrm{/}2}\mathrm{.}$ Any missing or unverified panel-pair geometry, identity, derivative bound, arithmetic radius or certificate returns uncertified and no matrix-defect claim. If no valid ${\gamma }_h$ is supplied, return the assembled matrix and its entrywise certificates without a matrix-defect bound. In particular, the batch-26 Duffy rule alone does not certify arbitrary coincident or edge-adjacent pairs.

Termination of the standard branches, entrywise certificate correctness, and the bilinear operator-defect bound for a successful output are proved in A certified matrix defect transfers to a Galerkin error bound ↗. The procedure above specifies the actual branches, recorded error quantities and uncertified behavior.