Spectral Methods for First-Kind Equations on Nonspherical Geometries

Definitions and mathematical statement

Research problem. Prove stability and convergence of fully discrete spectral methods for first-kind equations on geometries other than the sphere. A first-kind operator equation has the form $\mathcal K_\Sigma u=f$, with $\Sigma$ the geometry under consideration; the record does not specify the operator, kernel, function spaces, or admissible geometries. A spectral method approximates the unknown in finite-dimensional spaces generated by global basis functions. Fully discrete means that the numerical realization, including the prescribed quadrature or other discretization of operator evaluations and data, is included in the analysis, rather than analyzing only an exactly assembled finite-dimensional projection.

Once a well-posed operator setting $\mathcal K_\Sigma:X_\Sigma\to Y_\Sigma$ and a particular discrete scheme have been fixed, write $\mathcal K_{\Sigma,N}u_N=f_N$ on its approximation spaces. The requested theory should establish discrete solvability, a stability estimate in stated compatible norms with a constant independent of the refinement parameter $N$ for the permitted discretizations, and convergence of $u_N$ to the exact solution as spectral resolution and quadrature accuracy increase under stated regularity and consistency hypotheses. It should make any dependence on $\Sigma$, regularity, and auxiliary discretization parameters explicit. No particular norm, convergence rate, geometry-uniform bound, or scheme is supplied by the source. In particular, stability of a first-kind equation must not silently mean a uniform inverse bound in an inappropriate fixed norm for a smoothing operator.

The supplied literature triage, checked 2026-08-21, labels the problem partially solved. It reports a solution for slender spheroids, while a general stability-and-convergence theorem for the broader nonspherical class was not found in the located literature. The exact slender-spheroid operator, scheme, and hypotheses are not reproduced, so this reported special case is not asserted for every first-kind equation or every spectral discretization.

Short English statement

Extend rigorous stability and convergence analysis of fully implemented spectral schemes for first-kind equations beyond spherical geometries. The analysis must include numerical operator assembly and use norms suited to the chosen equation. The supplied triage reports a slender-spheroid special case.

Sources

• Local UnsolvedMath registry, permanent ID 30000020, code catalogue entry; Spectral Methods for First-Kind Equations on Nonspherical Geometries, Oberwolfach Reports Open Problems.
• Geomathematik (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/27, pp. 1371--1448. https://doi.org/10.4171/owr/2004/27. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label Ganesh: https://doi.org/10.1016/S0377-0427(03)00488-6; title and detailed theorem statement absent.
• Supplied literature label official report: https://ems.press/content/serial-article-files/45921.
• Literature triage checked 2026-08-21, status partially_solved; record updated 2026-09-16. Verify this dated assessment and cited claims independently.