arXiv subjects

David Scott Winterrose

Publications and source records attributed to David Scott Winterrose.

4 recordsLinked to original sources

arXiv2026

High-order $L^2$-Galerkin schemes for the decoupled potential integral equations

We prove Sobolev space well-posedness of the decoupled potential integral equations. Using basic pseudo-differential techniques, discrete stability is obtained for the standard $L^2$-pairing, and we deduce spectral convergence for a high-order scheme employing discontinuous basis functions, or with additional element-wise tangential continuity imposed in the vectorial part of the system. The scheme is shown to be much better conditioned than a mature commercial EFIE/CFIE solver. In many cases appearing almost independent of wavenumber, mesh density, and the order.

math.NA↗

arXiv2024

A structure theorem for fundamental solutions of analytic multipliers in $\mathbb{R}^n$

Using a version of Hironaka's resolution of singularities for real-analytic functions, any elliptic multiplier $\mathrm{Op}(p)$ of order $d>0$, real-analytic near $p^{-1}(0)$, has a fundamental solution $μ_0$. We give an integral representation of $μ_0$ in terms of the resolutions supplied by Hironaka's theorem. This $μ_0$ is weakly approximated in $H^t_{\mathrm{loc}}(\mathbb{R}^n)$ for $t<d-\frac{n}{2}$ by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

math.AP↗

arXiv2023

Algebras of pseudo-differential operators acting on holomorphic Sobolev spaces

We search for pseudo-differential operators acting on holomorphic Sobolev spaces. The operators should mirror the standard Sobolev mapping property in the holomorphic analogues. The setting is a closed real-analytic Riemannian manifold, or Lie group with a bi-invariant metric, and the holomorphic Sobolev spaces are defined on a fixed Grauert tube about the core manifold. We find that every pseudo-differential operator in the commutant of the Laplacian is of this kind. Moreover, so are all the operators in the commutant of certain analytic pseudo-differential operators, but for more general tubes, provided that an old statement of Boutet de Monvel holds true generally. In the Lie group setting, we find even larger algebras, and characterize all their elliptic elements. These latter algebras are determined by global matrix-valued symbols.

math.AP↗

arXiv2022

A holomorphic mapping property of analytic pseudo-differential operators

We study the holomorphic extendibility of $\text{Op}(p)u$, when $p$ is an analytic symbol, and explicit information is available on the domains of holomorphic extendibility of both $p$ and $u$. By a contour deformation argument, we obtain a precise local estimate of the domain of holomorphy of $\text{Op}(p)u$ in terms of the information on $p$ and $u$.

math.AP↗