On optimal $$L^2$$L2- and surface flux convergence in FEM

We show that optimal $$L^2$$L2-convergence in the finite element method on quasi-uniform meshes can be achieved if, for some $$s_0 > 1/2$$s0>1/2, the boundary value problem has the mapping property $$H^{-1+s} \rightarrow H^{1+s}$$H-1+s→H1+s for $$s \in [0,s_0]$$s∈[0,s0]. The lack of full elliptic regularity in the dual problem has to be compensated by additional regularity of the exact solution. Furthermore, we analyze for a Dirichlet problem the approximation of the normal derivative on the boundary without convexity assumption on the domain. We show that (up to logarithmic factors) the optimal rate is obtained.