In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency of the error estimator. These results hold true for both nonlinear partial differential equations and nonlinear functionals of interest. Furthermore, the DWR method employed in this work accounts for balancing the discretization error with the nonlinear iteration error. We also perform careful studies of the remainder term that is usually neglected. Based on these theoretical investigations, several algorithms are designed. Our theoretical findings and algorithmic developments are substantiated with some numerical tests.

Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

In this work, we design a posteriori error estimation and mesh adaptivity for multiple goal functionals. Our method is based on a dual-weighted residual approach in which localization is achieved in a variational form using a partition-of-unity. The key advantage is that the method is simple to implement and backward integration by parts is not required. For treating multiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete error problem) and suggest an alternative way for its computation. Our algorithmic developments are substantiated for elliptic problems in terms of four different numerical tests that cover various types of challenges, such as singularities, different boundary conditions, and diverse goal functionals. Moreover, several computations with higher-order finite elements are performed.

In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized p-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. numerical example of a quasi-linear PDE constraint, and multiple goal-oriented a posteriori error estimation.In the following, we briefly refer to studies that treat parts of the three topics. Optimal control problems (specifically, a priori estimates and optimality conditions) with quasi-linear (as the p-Laplacian can be classified) elliptic PDE constraints were considered in [16,18,15]. More recently, the extension to optimal control with parabolic PDEs was discussed in [9] and [14].Optimal control problems with (single) goal functionals were investigated in [6,40,5,50,52,43]. The p-Laplacian and a posteriori error estimates were considered in [36,12,20,13], and, more specifically, for goal functional evaluations, we refer to [34,44,25]. To estimate goal functionals, we adopt the dual-weighted residual (DWR) method [7,8] in which an adjoint problem is solved to obtain (local) sensitivity measures that are used for mesh refinement. As is well-known, using a gradientbased approach for the numerical solution of optimal control problems, the same adjoint problem as for the DWR error estimator can be employed. For this reason, it is natural to combine gradient-based optimization with adjoint-based error estimation.We are specifically interested in an extended DWR version in which the discretization and (linear/nonlinear) iteration error are balanced [39,44,37]. As localization technique we employ integration by parts as done in [8] or, for residual based error estimates, in [49]. The extension of [44] to multiple goal functionals was recently undertaken in [25].Three major aims constitute the main contents of this paper: first, the design of a framework for goal-oriented error estimation for optimal control subject to a nonlinear PDE and balancing the discretization and nonlinear iteration error (Section 3). From the optimization point of view, we carefully revisit the important elements for the DWR estimator for optimization problems. The main result in this respect is the a posteriori error representation for the reduced optimal control system for an abstract problem formulation. The second aim is the extension to the simultaneous control of multiple goal functionals (Section 4). As a third goal, based on our theoretical developments, we carefully design an adaptive soluti...