# Towards new algorithms and discretization

Doing pioneer work in science and answering questions of the future—these are tasks of young researchers that the European Research Council (ERC) supports with ERC Starting Grants. This type of grant provides up to 1.5 million euros for five years. Prof. Dr Dietmar Gallistl from Friedrich Schiller University Jena (Germany) is one of the researchers selected for funding this year. Within the framework of his project—‘Discretization and adaptive approximation of fully nonlinear equations’ (DAFNE)—, the mathematician intends to explore new numerical methods for a class of differential equations in order to expand their potential for possible applications.

“I am delighted to receive the grant, because over the next five years it will give me the time, freedom and flexibility to implement such a research project—together with a young and creative team,” says Dietmar Gallistl from the University of Jena. He wants to use the third-party funding of 1.45 million euros to finance two postdoc positions and two doctorral candidates. “On the one hand, the ERC Starting Grant confirms once again that Friedrich Schiller University has gained an excellent young researcher. On the other hand, such funding always rewards project ideas that could become promising research foci at our University in the next years”, says Prof. Dr Georg Pohnert, Vice-President for Research at the University of Jena, acknowledging the success of his colleague who has been conducting research in Jena since last winter semester.

Dividing the continuum into intervals

In the years to come, Dietmar Gallistl—who primarily conducts basic research in the field of numerical analysis—will explore how the ‘finite element method’ can be applied to the class of fully nonlinear equations. “In numerical analysis, we find ways to not only describe equations and analyse their nature, but to actually solve them approximately—and as efficiently as possible,” explains the 33-year-old scientist. “For this purpose, we generally use algorithms, nowadays usually applied by means of modern computing technology.” But even such computers reach their limits in terms of computing power, making it necessary to subdivide continuous mathematical problems—i. e. those involving an infinite number of numbers—into manageable smaller parts and thus approximate the solution of an equation as closely as possible. This is achieved by a process called discretization. One method used for that purpose is the finite element method, where a body, for example, is subdivided into many small parts. “It is frequently applied in engineering, for example when calculating the deformation of elastic solids in civil engineering,” explains the mathematician of the University of Jena.

Low effort, high approximate value

The discretization method may possibly be transferred to other classes of equations—e. g. fully nonlinear equations. These are more likely to be used in subject areas where problems are not modelled physically or mechanically—e. g. in financial mathematics or geometry. So far, little research has been done on adaptive methods such as the finite element method for this class of equations—and this is exactly what Dietmar Gallistl wants to change: “My goal is to find out how to use finite elements for certain fully nonlinear equations through certain regularizations, initially in two dimensions; the computational effort should be as low as possible and the approximation to the solution as exact as possible.” Possibly, this basic research could lay the foundation for new algorithms in the application areas of fully nonlinear equations.

**(Deutsch) Wissenschaftlicher Ansprechpartner:**

Prof. Dr Dietmar Gallistl

Institute of Mathematics, Friedrich Schiller University Jena

Ernst-Abbe-Platz 2, 07743 Jena, Germany

Telephone: +49 3641 946141

Email: dietmar.gallistl[at]uni-jena.de