2 research outputs found
Multigoal-oriented optimal control problems with nonlinear PDE constraints
In this work, we consider an optimal control problem subject to a nonlinear
PDE constraint and apply it to the regularized -Laplace equation. To this
end, a reduced unconstrained optimization problem in terms of the control
variable is formulated. Based on the reduced approach, we then derive an a
posteriori error representation and mesh adaptivity for multiple quantities of
interest. All quantities are combined to one, and then the dual-weighted
residual (DWR) method is applied to this combined functional. Furthermore, the
estimator allows for balancing the discretization error and the nonlinear
iteration error. These developments allow us to formulate an adaptive solution
strategy, which is finally substantiated via several numerical examples
Some a Posteriori Error Estimators for p-Laplacian Based on Residual Estimation or Gradient Recovery
n this paper, we first derive a posteriori error estimators of residual type for the finite element approximation of the p-Laplacian, and show that they are reliable, and efficient up to higher order terms. We then construct some a posteriori error estimators based on gradient recovery. We further compare the two types of a posteriori error estimators. It is found that there exist some relationships between the two types of estimators, which are similar to those held in the case of the Laplacian. It is shown that the a posteriori error estimators based on gradient recovery are equivalent to the discretization error in a quasi-norm provided the solution is sufficiently smooth and mesh is uniform. Under stronger conditions, superconvergnece properties have been established for the used gradient recovery operator, and then some of the gradient recovery based estimates are further shown to be asymptotically exact to the discretization error in a quasi-norm. Numerical results demonstrating these a posteriori estimates are also presented