Proper Generalized Decomposition (PGD) methods are highly efficient for high-dimensional problems but are traditionally limited to geometrically separable domains. In this talk, we present a reduced-order framework that circumvents this geometric limitation by coupling PGD with Nitsche’s method, targeting the solution of the Poisson equation on L-shaped domains. First, the L-shaped domain is embedded into a fictitious separable Cartesian product domain to maintain the tensor-product structure required by PGD. Nitsche’s method is then employed to impose boundary conditions along the re-entrant edges, avoiding conformal meshing or Lagrangian multipliers. The solution field is approximated by a finite sum of variable-separated modes, computed via an alternating directions algorithm. To address the numerical challenges associated with corner singularities and high-order mode stagnation, a randomized initialization strategy combined with an adaptive energy-based truncation criterion is introduced. This ensures that the solver captures the dominant physical behavior while filtering out numerical noise. Numerical examples are presented to validate the accuracy, efficiency, and robustness of the proposed approach against high-fidelity finite difference reference solutions. The results show that the PGD-Nitsche coupling provides an effective and scalable solution for non-separable geometries, achieving high accuracy with a compact modal basis.