A posteriori error analysis, Pod-Deim reduced order geometrically parametrized models and unfitted FEMs

TL;DR

This paper develops a comprehensive a posteriori error analysis framework for Pod-Deim reduced models combined with unfitted FEMs on parametrized domains, supported by theoretical proofs and numerical validation.

Contribution

It introduces new error estimators for Pod-Deim reduced models with unfitted FEMs, including theoretical analysis and practical validation, addressing ghost-penalty effects.

Findings

Residual estimators are effective and demonstrate exponential decay.

Ghost-penalty degrees of freedom inflate effectivity indices.

Numerical experiments confirm theoretical predictions and online speedup.

Abstract

We develop and analyze a posteriori error estimators for a proper orthogonal decomposition-discrete empirical interpolation method (Pod-Deim) reduced order model applied to a parametric Poisson equation posed on a parameter-dependent domain defined by a level-set function. The full-order discretisations employ a cut finite element method (Cutfem) with Nitsche boundary conditions and ghost-penalty stabilization. Three complementary estimators are proposed: (i) Deim approximation quality indicators for the stiffness matrix and force vector, which are constant in the number of Pod modes, (ii) dual-norm residual estimators in both plain and Jacobi-preconditioned form, and (iii) a Pod tail-energy indicator. A rigorous theoretical framework is established, comprising a uniform coercivity result for the Cutfem bilinear form, an active-dof residual bound that accounts for ghost-penalty degrees of freedom, a combined a posteriori bound, and sharp effectivity analysis for the residual estimators. The key theoretical finding is that the large observed effectivity indices are explained by ghost-penalty degree-of-freedom inflation, and that restricting the residual to active degrees of freedom is predicted to reduce effectivity. Numerical experiments on a parametric ellipse domain with semi-axes confirm the theoretical predictions, achieve significant online speedup, and demonstrate algebraic convergence of the true error alongside exponential decay of the residual estimators.