Domain Decomposition-Based Coupling of High-Fidelity Finite Element and Reduced Order Operator Inference Models Using the Schwarz Alternating Method

TL;DR

The paper addresses the high cost of high-fidelity FE analyses for multi-scale problems and robustness issues in data-driven ROMs. It introduces a hybrid domain decomposition framework that couples subdomain HFMs with OpInf ROMs through the Schwarz alternating method, enabling nonconformal meshes and online integration of HFM data. The OpInf-ROM training is performed nonintrusively, and the approach can operate without explicit regularization in the Schwarz context. Numerical experiments on a 2D convection-diffusion-reaction problem demonstrate that the OpInf-FE Schwarz coupling achieves accuracy comparable to a full FE model with significant CPU-time savings, outperforming monolithic OpInf models and reducing training-tuning needs. The method offers flexibility for extension to 3D and parallel Schwarz implementations, enhancing scalability for complex multiphysics simulations.

Abstract

We propose a novel hybrid domain decomposition method that couples sub-domain-local high-fidelity finite element (FE) models with reduced order models (ROMs) using the Schwarz alternating method. By integrating the noninstrusive Operator Inference (OpInf) ROM, our approach accelerates the Schwarz process while allowing for geometry and mesh flexibility. We demonstrate the effectiveness of the new OpInf-FE method on a convection-dominated convection-diffusion-reaction problem, achieving stable and accurate predictive solutions while improving the ROM training process.

Figures (1)

• Figure 1: CDR numerical solutions at $T = 5.0$