Super-localized orthogonal decomposition for convection-dominated diffusion problems

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:math> -norm, the Galerkin projection onto this generalized finite element space even yields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> -independent error bounds, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math> -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.