Workshop on Numerical Methods and Analysis in CFD - Abstract

Freese, Philip

Super-localized orthogonal decomposition for convection-dominated diffusion problems

In this talk, we present a multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The application of the solution operator to piece-wise constant right-hand sides on some arbitrary coarse mesh defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the L²-norm, the Galerkin projection onto this generalized finite element space even yields ε-independent error bounds.

We construct an approximate local basis that turns the approach into a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization, which we conjecture to decay super-exponentially, can be estimated in an a posteriori way. Numerical experiments indicate ε-independent convergence without preasymptotic effects, even in the under-resolved regime of large mesh Péclet numbers.