Super-localized orthogonal decomposition for convection-dominated diffusion problems.

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Partial Differential Equations

Background:

• Convection-dominated diffusion problems often exhibit sharp gradients, posing challenges for standard numerical methods.
• Large Péclet numbers indicate a dominance of convective transport over diffusive transport, leading to numerical instability.
• Existing multi-scale methods may struggle with robustness and pre-asymptotic effects in these regimes.

Purpose of the Study:

• To develop a novel multi-scale method for convection-dominated diffusion problems at large Péclet numbers.
• To establish error bounds independent of the singular perturbation parameter.
• To achieve robust convergence without pre-asymptotic effects.

Main Methods:

• Application of the solution operator to piecewise constant right-hand sides on a coarse mesh.
• Definition of a finite-dimensional coarse ansatz space with favorable approximation properties.
• Construction of an approximate local basis, creating a Super-Localized Orthogonal Decomposition (SLOD) inspired method.
• A posteriori error estimation for basis localization error.

Main Results:

• The Galerkin projection onto the generalized finite element space yields singular perturbation parameter-independent error bounds for certain norms.
• Numerical experiments demonstrate Péclet number-robust convergence.
• The method shows no pre-asymptotic effects, even in the under-resolved regime.

Conclusions:

• The proposed multi-scale method effectively handles convection-dominated diffusion problems with large Péclet numbers.
• The approach provides robust and parameter-independent error estimates.
• This novel method offers improved convergence properties compared to existing multi-scale techniques.