A Posteriori Error Analysis, Pod-Deim Reduced Order Geometrically Parametrized Models And Unfitted FEMs
This work addresses error analysis for computational efficiency in scientific computing, particularly for engineers and scientists using reduced order models, but it is incremental as it builds on existing POD-DEIM and CutFEM methods.
The paper tackles the problem of a posteriori error estimation for reduced order models in parametric PDEs on parameter-dependent domains, developing three complementary estimators and establishing a rigorous theoretical framework that explains large effectivity indices and achieves significant online speedup with algebraic convergence of the true error and exponential decay of residuals.
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.