Explicit synchronous partitioned scheme for coupled reduced order models based on composite reduced bases

TL;DR

This paper develops a partitioned scheme for coupled reduced order models and full order models, enabling independent subdomain solutions with guaranteed non-singular Schur complements, validated through numerical tests on an advection-diffusion problem.

Contribution

It introduces a novel composite reduced basis approach ensuring non-singular Schur complements for coupled ROM-FOM problems, facilitating explicit partitioned solutions.

Findings

The method guarantees non-singular Schur complements regardless of mesh or basis size.

Numerical tests confirm the scheme's effectiveness for advection-diffusion problems.

The approach enables independent subdomain solutions in coupled reduced models.

Abstract

This paper formulates, analyzes, and demonstrates numerically a method for the partitioned solution of coupled interface problems involving combinations of projection-based reduced order models (ROM) and/or full order methods (FOMs). The method builds on the partitioned scheme developed in [1], which starts from a well-posed formulation of the coupled interface problem and uses its dual Schur complement to obtain an approximation of the interface flux. Explicit time integration of this problem decouples its subdomain equations and enables their independent solution on each subdomain. Extension of this partitioned scheme to coupled ROM-ROM or ROM-FOM problems required formulations with non-singular Schur complements. To obtain these problems, we project a well-posed coupled FOM-FOM problem onto a composite reduced basis comprising separate sets of basis vectors for the interface and interior variables, and use the interface reduced basis as a Lagrange multiplier. Our analysis confirms that the resulting coupled ROM-ROM and ROM-FOM problems have provably non-singular Schur complements, independent of the mesh size and the reduced basis size. In the ROM-FOM case, analysis shows that one can also use the interface FOM space as a Lagrange multiplier. We illustrate the theoretical and computational properties of the partitioned scheme through reproductive and predictive tests for a model advection-diffusion transmission problem.