We devise a novel framework for the error analysis of nonconforming finite element techniques with low-regularity solutions. The key is to use some recently-derived mollification operators that commute with the usual derivative operators. We show how to apply the approach in the context of Nitsche's boundary penalty method for a diffusion equation and for the time-harmonic Maxwell's equations. In both cases, we address the case of heterogeneous material properties. We also compare the derived error estimates to those obtained in the framework of the error analysis framework proposed by Gudi where a trimming operator is introduced to map discrete test functions into conforming test functions.