Some Remarks on the Optimal Error Estimates for the Finite Element Method on the L-Shaped Domain

In the a priori L² error analysis of the finite element method (FEM), the Aubin-Nitsche trick is often used. Usually, the convergence order of the L² error estimates by the Aubin-Nitsche trick is one order higher than the H₀¹ error estimates. As is well known, the convergence order obtained by this technique depends on the shape of the domain because it is dependent on the regularity of solutions for the associated dual problem on the same domain. In this paper, we introduce a technique for getting the optimal order L² error estimates on the L-shaped domain without Aubin-Nitsche trick. From the numerical evidence based on the guaranteed computations, we could still expect that such a domain dependency is not essential.