Julian STREITBERGER successfully defended his dissertation entitled "Optimal complexity of standard and goal-oriented adaptive FEM for general second-order linear elliptic PDEs" on 20 June 2024 and completed his doctoral studies in Technical Mathematics with distinction. Most problems in science and technology are described by differential equations for which a solution can usually not be given in closed form, even if an abstract mathematical proof guarantees that the solution exists and is unique. If one is not only interested in the qualitative behavior of the solution, but also in quantitative properties, numerical simulation is unavoidable. In his dissertation, Julian Streitberger formulates and analyzes adaptive algorithms for the numerical solution of general and, in particular, non-symmetric second-order linear elliptic partial differential equations (PDEs). The aim of the research project was to develop algorithms for which it is possible to mathematically guarantee that the error decays optimally with the computing time. While only recently the optimal solution of symmetric PDEs was mathematically substantiated, computational time-optimal algorithms for non-symmetric problems were still open. The corresponding research question is now answered in this excellent dissertation: After the finite element discretization, the resulting non-symmetric system of equations is symmetrized at the functional-analytic level by means of the so-called Zarantonello iteration — analogously to the state-of-the-art proof of the Lax-Milgram lemma. The resulting symmetric systems of equations are then solved iteratively (and inexactly) using an optimal multigrid method. Altogether, this results in a total of three nested loops in the adaptive algorithm: discretization – symmetrization – iterative solution.

The central contribution of Julian Streitberger is now to rigorously analyze, adaptively control and balance the computational costs (and thus the computing time) of these loops in order to finally be able to mathematically formulate and prove optimal convergence behavior. In total, the doctoral studies led to four scientific publications, which are summarized in the dissertation. The reviewers of the dissertation were Prof. Emmanuil H. GEORGOULIS (Heriot-Watt University Edinburgh & NTU Athens), Prof. Christian KREUZER (TU Dortmund University) and Prof. Dirk PRAETORIUS (TU Vienna). Dr. Julian Streitberger will initially remain at TU Wien for a few more months as a postdoc in the workgroup Numerics of PDEs headed by Prof. Dirk Praetorius.