Super-localized orthogonal decomposition for convection-dominated diffusion problems

• Published in: BIT Vol. 64; no. 3; p. 33
• Main Authors: Bonizzoni, Francesca, Freese, Philip, Peterseim, Daniel
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.09.2024; Springer Nature B.V
• Subjects: Asymptotic methods; Computational Mathematics and Numerical Analysis; Convection; Decomposition; Error analysis; Mathematics; Mathematics and Statistics; Multiscale analysis; Numeric Computing; Peclet number; Research Paper; Singular perturbation; 65N15; 65N12; 35B25; Multi-scale method; Convection-dominated diffusion; Numerical homogenization; Singularly perturbed; Super-localization; 65N30
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-024-01035-8

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L 2 -norm, the Galerkin projection onto this generalized finite element space even yields ε -independent error bounds, ε being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate ε -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.