A DEIM driven reduced basis method for the diffuse Stokes/Darcy model coupled at parametric phase-field interfaces

TL;DR

This paper introduces a reduced basis method using DEIM for efficiently solving coupled Stokes/Darcy equations with parametric, topology-changing geometries, ensuring positive-semidefiniteness and enabling large-scale subsurface damage analysis.

Contribution

It develops a novel DEIM-based reduced basis approach with non-negativity preservation for parametric phase-field interface problems in coupled Stokes/Darcy models.

Findings

Efficient simulation of large-scale, parametric Stokes/Darcy problems.

Guarantees positive-semidefiniteness of system matrices.

Successfully applied to subsurface damage characterization.

Abstract

In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly via a phase-field indicator function. In our reduced order model, we approximate the parameter-dependent phase-field function with a discrete empirical interpolation method (DEIM) that enables affine decomposition of the associated linear and bilinear forms. In addition, we introduce a modification of DEIM that leads to non-negativity preserving approximations, thus guaranteeing positive-semidefiniteness of the system matrix. We also present a strategy for determining the required number of DEIM modes for a given number of reduced basis functions. We couple reduced basis functions on neighboring patches to enable the efficient simulation of large-scale problems that consist of repetitive subdomains. We apply our reduced basis framework to efficiently solve the inverse problem of characterizing the subsurface damage state of a complete in-situ leach mining site.