Summary: Superconvergent discretization error estimates can be obtained when the solution is smooth enough and the finite element meshes enjoy some structural properties. The simplest one is that any two adjacent triangles form a parallelogram. Existing results on finite element estimates on superconvergent meshes are reviewed, which can be used for numerical analysis of Dirichlet control problems. Moreover, an error estimate is given for a variational normal derivative which is of higher order on superconvergence meshes. Graded meshes can be used as a remedy of the reduced convergence order in the case of quasi-uniform meshes when elliptic boundary value problems with singularities in the vicinity of corners are treated. Discretization error estimates on graded meshes are reviewed. Depending on the construction, graded meshes may or may not have superconvergence properties. The discretization error of an elliptic Dirichlet control problem is discussed in the case of superconvergent graded meshes. Results of a paper in preparation are announced, where error estimates for Dirichlet optimal control problems on superconvergent graded meshes will be shown.
For the entire collection see [Zbl 1422.65009].