Reduced basis solvers for unfitted methods on parameterized domains

TL;DR

This paper introduces a unified reduced basis framework for parametrized PDEs on complex, parameter-dependent domains, combining unfitted finite element methods with tensor-based techniques for efficient, accurate model reduction.

Contribution

It develops a deformation-based approach and localization procedures to handle geometric variability, extending reduced basis methods to unfitted and deformed configurations, including saddle-point problems.

Findings

Demonstrates high accuracy and efficiency on 2D and 3D problems

Successfully applies to Poisson, elasticity, Stokes, and Navier-Stokes equations

Extends reduced basis methods to complex geometries with stable saddle-point formulations

Abstract

In this paper, we present a unified framework for reduced basis approximations of parametrized partial differential equations defined on parameter-dependent domains. Our approach combines unfitted finite element methods with both classical and tensor-based reduced basis techniques -- particularly the tensor-train reduced basis method -- to enable efficient and accurate model reduction on general geometries. To address the challenge of reconciling geometric variability with fixed-dimensional snapshot representations, we adopt a deformation-based strategy that maps a reference configuration to each parameterized domain. Furthermore, we introduce a localization procedure to construct dictionaries of reduced subspaces and hyper-reduction approximations, which are obtained via matrix discrete empirical interpolation in our work. We extend the proposed framework to saddle-point problems by adapting the supremizer enrichment strategy to unfitted methods and deformed configurations, demonstrating that the supremizer operator can be defined on the reference configuration without loss of stability. Numerical experiments on two- and three-dimensional problems -- including Poisson, linear elasticity, incompressible Stokes and Navier-Stokes equations -- demonstrate the flexibility, accuracy and efficiency of the proposed methodology.