Asymptotically optimal lattice cubature formulas with a regular boundary layer in the space Hpμ(Ω)

• Published in: AIP conference proceedings Vol. 3004; no. 1
• Main Author: Jalolov, Ozodjon
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Melville American Institute of Physics 11.03.2024
• Subjects: Algorithms; Asymptotic properties; Boundary conditions; Boundary layers; Boundary value problems; Lower bounds; Operators (mathematics); Optimization; Partial differential equations; Upper bounds
• ISSN: 0094-243X; 1935-0465; 1551-7616; 1551-7616
• DOI: 10.1063/5.0199854

The so-called functional approach turned out to be very efficient in the study of various issues arising in the theory of approximate integration and partial differential equations and related branches of analysis. The essence of this approach (if we confine ourselves to the example of a boundary value problem for a differential equation) is that the differential equation with the boundary conditions is implemented as an operator acting in a specially selected functional space; the required information is obtained from the properties of this operator. S.L. Sobolev developed an algorithm for constructing cubature formulas, which he called formulas with a regular boundary layer. He proved the asymptotic optimality of these formulas and the upper-bound estimate of the norm of the error functional in space U2m (Ω), setting the principal term. The purpose of this study is to obtain a lower estimate (i.e., a lower bound) for any error functional of lattice cubature formulas for spaces Hpμ(Ω) and determine the asymptotical optimality of cubature formulas with a regular in the sense of Sobolev boundary layer in space Hpμ(Ω).