The hierarchical finite element method, using elements with variable aspect ratio, has proven to be useful for the numerical solution of PDEs and allows several concepts for the control of the adaptation process. In addition to the more conventional adaptive grids derived from L2- or H1-based error estimates, we study grids which are optimized with respect to the evaluation of linear functionals like the value of the solution at a fixed point. It is well-known that this requires the solution of a dual problem. As for the case of singular solutions, these grids are extremely refined at certain points, yielding different strategies for the solution vector and the right-hand side. This improves the order of the error with respect to L2- or H1-based adaptive grids, but causes additional difficulties for the design of efficient multigrid solvers.
This work has been supported bt the Bayerische Forschungsstiftung via FORTWIHR - The Bavarian Consortium for HPSC.