Multi-goal oriented a posteriori error estimates for nonlinear partial differential equations
Partial differential equations (PDEs) play an important role in many applications. However, the
solutions to these PDEs are known in rare cases. Thus, the partial differential equations must
be approximated by some discretization techniques like the Finite Element Method(FEM) leading to
In practice, often the solution is not of primary interest, but some goal functional (quantity of interest).
These quantities of interest are,e.g., mean value flux and point evaluations provided that they exist.
For instance, for the Navier-Stokes equations, these goal functionals could be pressure difference,
drag and lift. Moreover, there may be several quantities of interest in practical applications. In
this thesis,we extend the dual weighted residual method to multiple goal functionals at once in a
cost efficient way.
The dual weighted residual method gives information on the sensitivity of the goal functionals,
and allows us to estimate the error at the same time.
This leads to a posteriori error estimates for multiple goal functionals and to a corresponding
Adaptive Finite Element Method (AFEM) aiming at the accurate computation of all goal functionals
at once. Furthermore, we extend the methodology to optimal control problem subject to some
nonlinear PDE constraint.
Here, a reduced optimization problem is formulated in terms of the control variable. Moreover,
we derive two-sided bounds for the goal oriented error estimates for higher-order approximation
and interpolation. Using a saturation assumption, we derive lower bounds yielding the efficiency
of the error estimator. These results hold true in the case of nonlinear PDEs and nonlinear goal
functionals. We also perform careful studies of the remainder term that is usually neglected.
Based on these theoretical investigations,several algorithms are designed.