Least-Squares Recovery of Dual Variables in ROMs for Saddle-Point Problems

In this work, we introduce a methodology for recovering the reduced Lagrange multiplier in Reduced Order Models (ROMs) of general saddle-point problems. Specifically, the multiplier is determined as the least-squares minimizer of the residual of the reduced primal variable in a dual norm. We prove that this procedure yields a unique solution under the condition that the full-order primal and dual spaces satisfy the inf-sup stability condition. This is an extension to general saddle-point problems of the method recently introduced in [6] to recover the reduced pressure in ROMs of incompressible flows. We further show that the proposed approach is equivalent to solving the reduced mixed problem with an enriched reduced primal basis, augmented by the supremizers of the reduced multiplier. We establish optimal error estimates for the reduced multiplier applicable to general saddle-point problems. Numerical experiments for heat diffusion in composite media validate our theoretical findings and demonstrate the suitability of the method to solve engineering problems.