POD reduced order modeling for evolution equations utilizing arbitrary finite element discretizations

TL;DR

This paper introduces a POD-based reduced order modeling approach for nonlinear parabolic evolution equations that accommodates arbitrary finite element discretizations, enabling flexible and accurate model reduction without requiring snapshot interpolation.

Contribution

It develops a POD method using the eigensystem of the correlation matrix, allowing for snapshots in different finite element spaces without interpolation, enhancing flexibility in spatial discretization.

Findings

The method accurately estimates errors from spatial, temporal, and POD approximations.

Numerical tests demonstrate the approach's effectiveness with adaptive meshes.

The approach is applicable to complex systems like the Cahn-Hilliard equation.

Abstract

The main focus of the present work is the inclusion of spatial adaptivity for the snapshot computation in the offline phase of model order reduction utilizing Proper Orthogonal Decomposition (POD-MOR) for nonlinear parabolic evolution problems. We consider snapshots which live in different finite element spaces, which means in a fully discrete setting that the snapshots are vectors of different length. From a numerical point of view, this leads to the problem that the usual POD procedure which utilizes a singular value decomposition of the snapshot matrix, cannot be carried out. In order to overcome this problem, we here construct the POD model / basis using the eigensystem of the correlation matrix (snapshot gramian), which is motivated from a continuous perspective and is set up explicitly e.g. without the necessity of interpolating snapshots into a common finite element space. It is an advantage of this approach that the assembling of the matrix only requires the evaluation of inner products of snapshots in a common Hilbert space. This allows a great flexibility concerning the spatial discretization of the snapshots. The analysis for the error between the resulting POD solution and the true solution reveals that the accuracy of the reduced order solution can be estimated by the spatial and temporal discretization error as well as the POD error. Finally, to illustrate the feasibility our approach, we present a test case of the Cahn-Hilliard system utilizing h-adapted hierarchical meshes and two settings of a linear heat equation using nested and non-nested grids.