The Aubin--Nitsche Trick for Semilinear Problems
Provides a theoretical extension for error analysis in numerical methods for semilinear PDEs, but is incremental as it adapts an existing technique.
The authors generalize the Aubin–Nitsche trick, a tool for linear PDEs, to semilinear problems arising from minimization, enabling L²-error estimates for discretizations where previously not applicable.
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.