The Aubin--Nitsche Trick for Semilinear Problems

TL;DR

This paper extends the Aubin--Nitsche trick, traditionally used for linear PDEs, to a class of semi-linear problems, enabling $L^2$-error estimates for more complex quasilinear discretizations.

Contribution

The paper generalizes the Aubin--Nitsche trick to semi-linear PDEs, broadening its applicability beyond linear problems.

Findings

Successfully extended the Aubin--Nitsche trick to semi-linear problems.

Provides a new approach for $L^2$-error estimation in quasilinear discretizations.

Enhances error analysis techniques for non-linear PDE discretizations.

Abstract

The Aubin--Nitsche trick is a common tool to show $L^2$-error estimates for discretizations of $H^1$-elliptic linear partial differential equations arising for example as Euler--Lagrange equations of a quadratic energy functional. The technique itself is linear: for quasilinear problems it is not applicable. We generalize the Aubin--Nitsche trick to a class of minimization problems closely related to semi-linear partial differential equations.