Super-localized orthogonal decomposition for convection-dominated diffusion problems

Abstract

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 \(\varepsilon \)-independent error bounds, \(\varepsilon \) 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 \(\varepsilon \)-robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.