Numerical methods for physical problems with variable growth conditions

• Applicant: Dr. Alex Kaltenbach
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 525389262

The proposed research project aims at the theoretical and experimental investigation of numerical methods for physical models involving variable growth conditions. Physical models involving variable growth conditions are of interest in many engineering applications, including, e.g., smart materials like (micro-polar) electro-rheological fluids, chemically reacting fluids, and thermo-rheological fluids. Most of these models have a related structure, consisting of the generalized Navier–Stokes equations coupled to equations describing additional physical quantities, e.g., an electric field, a temperature field, or a concentration. The coupling inter alia takes place through a power-law index which depends on the physical quantity described by the additional equations and is, thus, time- and space-dependent. These models are also similar from a purely mathematical point of view; if the additional coupled equations and the described physical quantity are understood, then the remaining mathematical challenges in all models are the same: all models treat the generalized Navier–Stokes equations with a time- and space-dependent power-law index, the so-called (unsteady) p(t, x)-Navier–Stokes equations. Whereas initial (purely theoretical) numerical investigations have been carried out for the p(t, x)-Navier–Stokes equations in the steady case, there have been only a few investigations in the unsteady case. It is the purpose of this research project to fill this lacuna. More precisely, the envisaged research project comprises the following two mutually connected main objectives: Work Area A: In this part of the project, a finite element approximation of the steady p(x)-Navier–Stokes equations will be examined for weak convergence and error estimates. Numerical experiments will be carried out to confirm the theoretical findings. Work Area B: In this part of the project, a fully-discrete Rothe–Galerkin approximation (i.e., a back-ward Euler step in time and the finite element method in space) of the unsteady p(t,x)-Navier–Stokes equations will be examined for weak convergence. Numerical experiments will be carried out to confirm the theoretical findings.

• DFG Programme: WBP Fellowship
• International Connection: Italy
• Host: Professor Dr. Luigi Carlo Berselli