# Research

In general scientific computing is a field of study that uses advanced algorithms to solve complex problems. It involves the development and application of mathematical models and simulations to understand and solve scientific problems. Scientific computing is a very broad field that intersects with a variety of disciplines, including mathematics, science, engineering, and computer science. Here, at the Institute of Numerical and Applied Mathematics, we focus on the numerical and algorithmic aspects of scientific computing.

## Mayor research topics

### Stochastic elliptic partial differential equations

Stochastic Partial Differential Equations (SPDEs) are a type of mathematical model used in scientific computing to represent systems that are influenced by random effects. One of the main challenges with SPDEs is that they often lead to high-dimensional problems. This is because the solution to a SPDE is typically a random field, which is a function of both space and random parameters.

This is a chalanging scientific computing problem because the computational cost of solving high-dimensional problems can be prohibitively high. This is due to the fact that the number of degrees of freedom needed to discretize a high-dimensional space grows exponentially with the dimension. To avoid this adaptive methods that use a nearly optimal amount of degrees of freedom are being developed.

### Incompressible viscous flow problems

Incompressible viscous flow problems are typically governed by the Navier-Stokes equations, or related (simplyfied) PDEs. A variety of tools have been developed to address these problems efficiently. A wide variety of issues have to be considered, opften in a simplified setting. Therefor parabolic problems, fluid-liquid interaction and saddle-point problems are studied.

Numerous tools exist and are being developed further to tackle these problems. Different families of finite element methods can be used to preserve the underlying structure, such as continuity and divergence-constrained. Or conditions can be relaxed and reinterpreted to gain a computational advantage.