Guaranteed inf-sup bounds and existence verification for semilinear elliptic problems via nonconforming finite elements

Abstract

A Newton-Kantorovich-type argument enables the a posteriori existence verification of a locally unique regular root near a computed approximation. This framework allows for non-selfadjoint problems and extends the existing verification theory to nonconforming discretisations. A key ingredient is a guaranteed lower bound for the continuous inf-sup constant of the linearisation obtained with a novel approximation error bound for nonconforming schemes. All quantities are obtained from a postprocessing on the same discretisation within the adaptive loop. The theory is applied to a fourth-order formulation of the stationary 2D Navier-Stokes equations and illustrated by numerical experiments.