In brief we do mathematical modelling, mathematical an numerical analysis and computer simulations. The list of publications can be found here.

We strive for mathematically rigorous description of natural phenomena on all levels of the mathematical modelling – development of the mathematical model, study of its mathematical properties, design of suitable numerical methods and finally computer simulations. We think that the successful description of natural phenomena requires excellence on all levels, therefore the team is composed of the researchers working in fundamental research in mathematical modelling, mathematical analysis and numerical mathematics. The existence of the centre allows us to combine the state-of-art results in all these fields, and to achieve the optimal and well-founded description of given natural phenomena.

**Keywords:** mathematical modelling, non-newtonian fluids, partial differential equations, weak solution, function spaces, Sobolev and Orlicz spaces, numerical mathematics, discretization of partial differential equations, discontinuous Galerkin method, a posteriori error estimates, numerical linear algebra, Krylov subspace methods

## Mathematical modelling

Modern technologies allows one to manufacture materials that have on the macroscopic scale unique properties that can not be modelled using the traditional concepts such as the Navier-Stokes fluid or linearised elasticity. Such properties are however not seen only in artificial materials, some common materials such as body tissues or geomaterials can exhibit unusual behaviour. Moreover the well-known material models fail if the materials are subject to extreme mechanical or thermal loading.

The challenge for mathematical modelling is to develop mathematical models describing behaviour of such materials in such extreme processes. This is not an academic issue since in many applications – biophysics, geophysics, medicine, material science to name a few – one really needs to face this challenge. In many situations it is impossible to use “microscopic” theories such as quantum mechanics to develop the models, since such models are either too complex, inapplicable or do not exist at all. Here one has to rely on **phenomenological theories** based on continuum mechanics and thermodynamics – this is precisely what the modelling group is focused on.

Our approach to modelling the **behaviour of complex materials in extreme processes** is based on the following concepts:

- maximization of the rate of the entropy production,
- natural configuration,
- implicit constitutive relations.

These three recently developed concepts allows one to develop models that are consistent with the laws of thermodynamics, and that can overcome the restriction of some classical theories such as the classical non-equilibrium thermodynamics or generalize the results of some recent theories such as the extenden irreversible thermodynamics. The three concepts have been shown to be applicable in describing the behaviour of polymer melts undergoing phase transitions, chemically reacting viscoelastic materials, blood clot formation (including biochemical reactions and their interplay with the mechanical processes) and so on.

Our research strategy is to investigate the features, limitations and possible generalizations of these concepts.

## Mathematical analysis

Form the mathematical point of view the models in continuum mechanics and thermodynamics lead to systems of nonlinear partial differential equations and represent a challenge for mathematical analysis.

We think that the understanding of the physical background of the models can help us in introducing the right concept of the solution to the given system of partial differential equations and in analysing the qualitative properties of such solution. In the light of results in continuum thermodynamics, the appropriate concept of the solution is the so-called **weak solution** or its variants (entropic solution, renormalized solution, suitable weak solution). The nonlinearity of the studied equations and the non-standard boundary conditions lead naturally to the need to use non-standard function spaces (generalized Orlicz and Sobolev-Orlicz spaces) in which one seeks the solution and that are worth of studying *per se*. The concept of solution then provides a solid foundation for construction of the optimal numerical methods for approximate solution of the given system of partial differential equations.

The awareness of physical background of the systems of nonlinear partial differential equations also allows us – after the dimensional analysis – formulate and proof conjectures on the **asymptotic behaviour** of these systems, and consequently identify the simplified systems that still describe the given phenomena, but are substantially more accessible for numerical simulations. (Long time behaviour or behaviour for vanishing coefficients such as low Reynolds number limits and so on.)

On the technical level, we focus on the **synthesis of the “steady” and “evolutionary” methods** in analysis of systems of nonlinear partial differential equations. By “steady” methods we mean the methods used in the calculus of variations, such as the Γ-convergence and related concepts in the theory of homogenization. By “evolutionary” methods we mean the methods for study of evolutionary partial differential equations (energy method, theory of monotone operators, gradient flows, viscosity solutions, transport theory). An important research direction is to study physically relevant problems where needs to face the problem of the changing character (parabolic to hyperbolic) of the leading differential operator in the system.

## Numerical mathematics

The ultimate goal is computer simulation of behaviour of complex materials in extreme processes. It requires the **design, analysis and implementation of numerical methods** that are:

- accurate and reliable (the simulation results must be accompanied with fully computable a posteriori error bounds),
- efficient (the computational cost must allow us to go beyond existing horizons in order to extract qualitatively new information),
- robust (the new tools must be applicable to a sufficiently large class of problems, which will justify the effort needed for their development).

Here we focus on the important and up to now relatively rarely studied **algebraic errors** and their inclusion in a posteriori error estimates. The study of the algebraic errors is, in our opinion, the fundamental research challenge in the field of numerical mathematics for the following years. The error analysis however can not be done without an in-depth knowledge of the results for the given systems of partial differential equations – this is the reason why the centre includes both mathematical analysis research group and numerical mathematics research group.

Concerning the techniques for the discretization of the systems of partial differential equations, our method of choice is the **discontinuous Galerkin method**. We are also interested in multiscale FEM and space-time FEM. For these methods we study convergence, stability and error estimates in the above sense. Further, the methods are studied with respect to the physical properties of the system (fulfilment of the discrete maximum principle, positivity of the solution).

On the discrete level (nonlinear algebraic equations) the solvers are often based on Newton or Broyden type methods. In many cases the linearised operator is not given explicitly but only via a procedure for matrix-vector multiplication. In this case we say that the method is **matrix-free** and solving such problems is a challenging task. We also consider another challenge in this field, namely the re-use of algebraic information from sequences of linearised systems. Moreover, the complex nonlinear problems often reflect hierarchical character which can be further exploited in saddle-point solvers or even in the framework of general multilevel methods.

Last but not least we focus on **Krylov subspace methods** for solving systems of linear algebraic equations, and on the links of these methods to the other fields in mathematics (model reduction). Here we build on the results obtained within the project Theory of Krylov subspace methods and its relationship to other mathematical disciplines.

The numerical methods we develop are implemented using the modern academic numerical libraries with regard to the possibility of using parallel computing architectures.