POD-Galerkin reduced-order modeling with adaptive finite element snapshots

TL;DR

This paper develops a POD-Galerkin reduced-order modeling approach for parametrized PDEs with snapshots computed via adaptive finite elements, avoiding the need for snapshot interpolation onto a common mesh.

Contribution

It introduces a method to construct POD-Galerkin models directly from adaptive finite element snapshots without interpolation, and analyzes the impact on error assessment.

Findings

Effective reduced-order models for linear elliptic problems.

Application to nonlinear time-dependent problems like Burgers equation.

Analysis of error implications due to adaptive snapshot spaces.

Abstract

We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection-diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear time-dependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.