A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems

TL;DR

This paper introduces a combined reduced basis and super-localized orthogonal decomposition method to efficiently solve parameter-dependent reaction-convection-diffusion problems with complex, heterogeneous coefficients at multiple scales.

Contribution

It presents a novel integration of RB and SLOD techniques to accelerate basis computation and compress solution operators for complex multiscale PDEs.

Findings

Significantly reduces computational cost for parametric multiscale problems.

Enables efficient treatment of multiple right-hand sides with minimal coarse solves.

Handles arbitrary rough and non-affine coefficients effectively.

Abstract

This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side.