# Multigoal-oriented error estimates for non-linear problems

• Bernhard Endtmayer , Ulrich Langer and Thomas Wick

## 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.

JEL Classification: 65N30; 65M60; 65J15; 49M15; 35Q74

## Acknowledgment

This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems’. The first author thanks the Doctoral Program on Computational Mathematics at JKU Linz the Upper Austrian Goverment for the support when starting the preparation of this work. The third author was supported by the Doctoral Program on Computational Mathematics during his visit at the Johannes Kepler University Linz in March 2018.

1. Funding: This work has been supported by the Austrian Science Fund (FWF) under the grant P 29181.

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