Construction of local reduced spaces for Friedrichs' systems via randomized training

TL;DR

This paper extends localized training methods to Friedrichs' systems, proving compactness of transfer operators and demonstrating effectiveness through numerical experiments on heterogeneous diffusion problems.

Contribution

It introduces a novel localized training approach for Friedrichs' operators, including theoretical compactness results and practical numerical validation.

Findings

Proved compactness of transfer operators for Friedrichs' systems.

Established energy decay and compactness in solution graph-spaces.

Numerical results show improved performance on heterogeneous diffusion problems.

Abstract

This contribution extends the localized training approach, traditionally employed for multiscale problems and parameterized partial differential equations (PDEs) featuring locally heterogeneous coefficients, to the class of linear, positive symmetric operators, known as Friedrichs' operators. Considering a local subdomain with corresponding oversampling domain we prove the compactness of the transfer operator which maps boundary data to solutions on the interior domain. While a Caccioppoli-inequality quantifying the energy decay to the interior holds true for all Friedrichs' systems, showing a compactness result for the graph-spaces hosting the solution is additionally necessary. We discuss the mixed formulation of a convection-diffusion-reaction problem where the necessary compactness result is obtained by the Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario involving heterogeneous diffusion fields with multiple high-conductivity channels, demonstrate the effectiveness of the proposed method.