M. Innerberger:

"Reliable goal-oriented adaptive FEM";

Supervisor, Reviewer: D. Praetorius, R. Becker, R. Stevenson; Institute of Analysis and Scientific Computing, 2022; oral examination: 2022-04-27.

This thesis considers goal-oriented adaptive finite element methods (GOAFEM), which aim to

approximate some quantity of interest, the goal value, derived from the solution of a partial

differential equation (PDE). Despite the practical relevance of GOAFEM, mathematical research

on it is scarce and, in particular, existing research on optimal algorithms for GOAFEM is

essentially limited to linear elliptic PDEs with linear goals. In this thesis we extend existing

results of GOAFEM towards practically more relevant cases and design algorithms that reliably

approximate the goal value at high (or even optimal) efficiency.

First, we give a brief overview of the existing results on optimal GOAFEM before we consider, for

the first time, GOAFEM for linear elliptic PDEs with quadratic goal. We propose an adaptive

algorithm that uses a linearized dual problem for error estimation and marking, and deals with

the arising linearization error. We prove convergence of this algorithm for every quadratic goal

and even convergence with optimal algebraic rates in the case that the Fréchet derivative of the

goal is compact.

Next, we investigate optimality results of GOAFEM for linear elliptic PDEs with linear goal, where

the primal and dual problem are solved by an (inexact) iterative solver. We observe that the

discrete goal value needs to be corrected in this case, and prove (linear) convergence of the

corrected goal error under the sole assumption that the solver is contractive. Furthermore, we

present criteria to stop the iterative solver on each step of the adaptive algorithm based on a

posteriori error estimates of both discretization error and algebraic error. If the involved

parameters are sufficiently small, the resulting adaptive algorithm is optimal with respect to the

number of degrees of freedom and even with respect to the total computational cost, which also

includes the cost of the iterative solver.

As an application of GOAFEM, we then consider parameter estimation problems for linear elliptic

PDEs that depend on a finite number of parameters. These parameters are inferred by

comparing experimental data to numerical simulations, which, by regarding the parameters as a

goal value, can be performed by GOAFEM. We prove a novel a priori estimate for the error in the

parameters, based on a set of PDEs corresponding to primal and dual problem. Using this

estimate as the basis of an estimate by usual a posteriori residual estimators for the energy

error, we are able to design an adaptive algorithm, where the a posteriori bound matches the

rate of convergence of the parameter error.

Throughout, we give numerical evidence to support our theoretical findings. The last part of this

thesis is dedicated to implementational aspects of numerical experiments for GOAFEM, where

we give the details of an object oriented implementation in Matlab. The library implements

higher-order FEM for second-order elliptic PDEs where the coefficients can be quite general,

covering also most cases typically arising from the iterative linearization of nonlinear PDEs. In

particular, the code covers all presented numerical experiments.

https://www.asc.tuwien.ac.at/praetorius/download/thesis/phd/innerberger2022.pdf

Created from the Publication Database of the Vienna University of Technology.