Computational nonlinear PDEs

01.09.2020–31.08.2025
FWF Research project
Principle investigator: Prof. Dirk PRAETORIOUS (E101)

Many problems in engineering and natural sciences are described by so-called partial differential equations. For many differential equations, it can be mathematically proven that a unique solution exists, but for very few differential equations can the solution be written down as an explicit formula. Therefore, the quantitative evaluation of the solution usually requires numerical methods that approximate the unknown solution as accurately as possible. Several sources of error must be considered in the overall error: The discretization error measures the unavoidable error that arises from replacing an infinite-dimensional function space (in which the exact solution lies) with a finite-dimensional finite element space. The linearization error measures the error that arises from transforming nonlinear equations into a sequence of linear problems, for example, using the Newton method. Finally, the linear systems of equations that arise in practice are often very large and are therefore not solved exactly, but rather by iterative algebraic methods, which introduces additional algebraic errors.

In the FWF-funded research project Computational nonlinear PDEs (https://www.doi.org/10.55776/P33216, opens an external URL in a new window), mathematical tools and adaptive algorithms are being developed that reliably control and balance discretization errors, linearization errors, and algebraic errors, such that the overall error is minimized in terms of computational effort and time. This ensures that the numerical simulation calculates an approximation in minimal time, where the error to the (unknown) exact solution of the differential equation does not exceed a specified tolerance. In our modern, highly technological world, such optimal algorithms are relevant because energy consumption of the computers generally scale with computational effort and time.

Project team members: Philipp BRINGMANN, Maximilian BRUNNER, Ani MIRACI, Julian STREITBERGER