Quasi-optimal nonconforming methods for second-order problems on domains with non-Lipschitz boundary

We introduce new nonconforming finite element methods for elliptic problems of second order. In contrast to previous work, we consider mixed boundary conditions and the domain does not have to lie on one side of its boundary. Each method is quasi-optimal in a piecewise energy norm, thanks to the discretization of the load functional with a moment-preserving smoothing operator.