**Prof. Michael Feischl is investigating how to achieve maximum precision in complicated computing tasks with minimal computer effort. He has been awarded an ERC Consolidator Grant for his work.**

When the world is simulated on a computer, the simulation is never completely accurate. Just as every image on the computer consists of individual pixels, the world is divided into small portions in order to do complicated scientific calculations. The way this is done has a decisive influence on how long the calculation takes and how precise the result is.

Prof. Michael Feischl from the Institute of Analysis and Scientific Computing at TU Wien develops mathematical methods to solve partial differential equations on the computer as quickly and precisely as possible. He has now been awarded an ERC Consolidator Grant by the European Research Council (ERC) - one of the most prestigious and highly endowed grants in the European research landscape.

## How precise can it be?

Suppose you want to calculate how the temperature in Austria will change over the next few days - how can this be described on the computer? Imagine a three-dimensional grid and store a temperature value (and other values such as pressure, wind speed, etc.) for each grid point. How the temperature will change at a particular point depends on the data of the surrounding points.It is often difficult to say how many grid points you need at which location: In stable weather conditions, the temperature is often fairly uniform over many kilometers and therefore a coarse grid is sufficient. In a thunderstorm, on the other hand, the temperature may change locally very quickly and an extremely fine grid is needed. The same applies to time: for fast processes, a high temporal resolution is required, for slower processes a lower resolution is sufficient.

## The core of the physical sciences

"We deal with partial differential equations," says Michael Feischl. "They are the basis of almost every physical computer simulation - from geophysics to astrophysics, from fluid mechanics to electromagnetism. Partial differential equations are also used in other areas such as financial mathematics or machine learning."These partial differential equations are solved on a kind of grid, but this grid does not have to be completely uniform. Perhaps you need better resolution in one area than elsewhere, perhaps not all time periods are equally critical. "If you want to increase precision and save computing time at the same time, you have to adapt the grids to the problem in question in an intelligent way," says Michael Feischl. "However, adaptive grid fitting for partial differential equations is still not well understood mathematically." For complex simulations in physics and technology, there are still no algorithms that can guarantee the optimal balance of precision and speed.

This is exactly what Michael Feischl wants to change in his ERC grant project. "The basis for this is a new mathematical finding that now, for the first time in the history of computational mathematics, opens up a way to solve the problem for realistic, time-dependent computational tasks," says Michael Feischl. The project combines several areas of mathematics - including finite element theory, matrix theory, non-linear partial differential equations and error estimation.