Abstract
We show that optimal
L
2
-convergence in the finite element method on
quasi-uniform meshes can be achieved if the underlying boundary value problem
admits a shift theorem by more than
1
/
2
. For this, 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.
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