A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement

We present a reduced basis method for parametrized partial differential equations certified by a dual-norm bound of the residual computed not in the typical finite-element “truth” space but rather in an infinite-dimensional function space. The bound builds on a finite element method and an associated reduced-basis approximation derived from a minimum-residual mixed formulation. The offline stage combines a spatial mesh adaptation for finite elements and a greedy parameter sampling strategy for reduced bases to yield a reliable online system in an efficient manner; the online stage provides the solution and the associated dual-norm bound of the residual for any parameter value in complexity independent of the finite element resolution. We assess the effectiveness of the approach for a parametrized reaction-diffusion equation and a parametrized advection-diffusion equation with a corner singularity; not only does the residual bound provide reliable certificates for the solutions, the associated mesh adaptivity significantly reduces the offline computational cost for the reduced-basis generation and the greedy parameter sampling ensures quasi-optimal online complexity.