Bernoulli-root sampling for optimal-order isogeometric least-squares collocation
Year: 2026
Type: article
Abstract: We present a Bernoulli-root least-squares collocation method for isogeometric discretizations of one-dimensional nondivergence-form model problems with a variable leading coefficient. On each knot span, the strong residual is sampled at two symmetric roots of the even Bernoulli polynomial \(B_{m(p)}\), where \(m(p)=2\lfloor p/2\rfloor\); consequently, consecutive degree pairs share the same sampling points. Although these sampling points coincide with the so-called spline-qualocation nodes, i.e., \textit{the quadrature-modified collocation nodes}, the proposed degree-paired selection rule is derived analytically from the cancellation of the leading Bernoulli-Peano defect rather than through numerical optimization. For every fixed spline degree \(p \geq 3\), whether even or odd, on uniform meshes and under the stated coefficient assumptions for \(u\in H^{p+3}(\Omega)\), we prove the optimal estimates \(\|u-u_{h,p}\|_{L^2(\Omega)}=O(h^{p+1})\) and \(\|u-u_{h,p}\|_{H^1(\Omega)}=O(h^p)\). The sampled least-squares form is shown to be stable on quasi-uniform meshes, and the corresponding \(H^2\) estimate of order \(O(h^{p-1})\) requires only \(u\in H^{p+1}(\Omega)\). The nonperiodic lower-norm analysis combines a boundary-compatible dual spline, maximum-norm stability of a weighted sampled projection, and a signed Bernoulli--Peano expansion. Numerical experiments confirm the theoretical convergence rates and illustrate their sensitivity to point placement. A tensor-product point rule is also provided for implementation, although the theoretical analysis remains one-dimensional.
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,830
OpenAlex ID: W7215124626