Iterative Methods for Model Reduction by Domain Decomposition

TL;DR

This paper introduces a domain decomposition approach combining full discretization and reduced order modeling to efficiently solve complex PDEs, demonstrated on nonlinear problems like the Laplace and Euler equations.

Contribution

It presents novel coupling strategies for integrating reduced order models with traditional discretizations in domain decomposition methods.

Findings

Reduced computational effort demonstrated on nonlinear PDEs

Effective coupling approaches improve solution efficiency

Numerical examples validate the method's applicability

Abstract

We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the two subdomains. In particular, in one subdomain we discretize the governing equations by a canonical scheme, whereas in the other one we solve a reduced order model of the original problem. Different approaches to couple the low-order model to the usual discretization are presented. The effectiveness of these approaches is tested on numerical examples pertinent to non-linear model problems including the Laplace equation with non-linear boundary conditions and the compressible Euler equations.