The Aubin--Nitsche Trick for Semilinear Problems

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.