Doctor's Theses (authored and supervised):
"Reliable goal-oriented adaptive FEM";
Supervisor, Reviewer: D. Praetorius, R. Becker, R. Stevenson;
Institute of Analysis and Scientific Computing,
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.
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.