A penalty free non-symmetric Nitsche type method for the weak imposition of boundary conditions

Abstract: In this note we show that the non-symmetric version of the classical Nitsche's method for the weak imposition of boundary conditions is stable without penalty term. We prove optimal $H<sup>1$-error estimates and $L<sup>2$-estimates that are suboptimal with half an order in $h$. Both the pure diffusion and the convection--diffusion problems are discussed.