An Aubin-Nitsche Lemma for a positivity-preserving finite element method

In this work we prove an Aubin-Nitsche Lemma for a positivity-preserving discretisation of an elliptic problem. Due to the nonlinearity of the discretisation, the result requires as a first step the proposal of a linearised adjoint problem that can be linked to the method of choice by appropriately selecting weights. This linearised adjoint problem, together with a regularity result, allow the proof of an optimal-order error estimate in the $L^2$-norm of the error.