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

TL;DR

This paper develops a framework for verifying the existence and bounds of solutions to semilinear elliptic problems using nonconforming finite elements, with applications to Navier--Stokes equations.

Contribution

It extends verification theory to nonconforming discretisations and provides guaranteed bounds using a novel a priori error estimator.

Findings

Guaranteed lower bounds on inf-sup constants obtained from a single discretisation

Extension of verification methods to non-selfadjoint and nonconforming problems

Numerical experiments demonstrate the effectiveness of the approach

Abstract

A Newton--Kantorovich-type argument enables the a posteriori existence verification of a unique regular root near a computed approximation, purely from computable data. This framework allows for non-selfadjoint problems and extends the existing verification theory to nonconforming discretisations. A key ingredient is a guaranteed lower bound on the continuous inf-sup constant from a quasi-optimal nonconforming discretisation that enables a novel a priori error estimator. All quantities are obtained by post-processing a single discretisation; convergence rates are proved. The theory is applied to a fourth-order formulation of the stationary two-dimensional Navier--Stokes equations and illustrated by numerical experiments.