Title: Convection-diffusion-reaction type equations with applications to engineering and life sciences

Summary

The seminar consists of presenting two projects: the modelling of biofilm growth in a raw water treatment process and the numerical analysis of an HDG method for fluid mechanics problems with smooth interfaces.

A Slow Sand Filter (SSF) consists of saturated packed sand with supernatant water on top through which water flows by gravity. SSFs are used in the production of drinking water due to their relatively simple and cheap construction, in addition to the fact that using members of microbial communities already present in raw water creates an effective biofilter for the removal of pathogens. A drawback of SSFs is that difficult sampling and complex microbial dynamics pose an obstacle for analysis and predictions. In order to overcome this, the biofilm growth in SSFs is modelled using a system of non-linear balance laws of convection-diffusion-reaction type with discontinuous flux for the concentration of species in the biofilm matrix and the flowing liquid suspension. Additionally, a Cahn-Hilliard type equation (fourth order non-linear PDE modelling phase separation) is used to define a constitutive equation for the convective velocities. The main aim of the first project is partly to obtain a comprehensive mathematical-ecological model formulated as a system of PDEs with a spatially discontinuous velocity, and partly to create a numerical method to approximate the solution.

The family of HDG methods is a type of finite element methods that use discontinuous basis functions constructed from local finite element spaces on each element, introducing an approximation of the trace of the solution on each inter-element boundary, and the prescription of a numerical flux across these. When dealing with curved domains, it is possible to use simplicial elements in conjunction with a transferring technique involving the extrapolation of the approximated gradient to account for the possible variational crime committed. It is possible to extend this technique to the case of interface problems, and it is in this context that the second project aims to develop the numerical analysis and implementation of such HDG methods for diffusion and convection-diffusion equations such as the Darcy and Oseen equations, coupling both in two sub-domains separated by a smooth interface.

The first part of the talk is based on the following manuscripts:

• Manríquez, J., Rosenqvist, T., Chan, S., Paul, C. J., Diehl, S., A mathematical-ecological model of biofilm growth in slow sand filters, manuscript in preparation (2024).
• Diehl, S., Manríquez, J., Paul, C. J., Rosenqvist, T., A simulation model for a convection-diffusion-reaction system with discontinuous flux modelling biofilm growth in slow sand filters, manuscript in preparation (2024).

The second part of the talk is based on the following publication and manuscript:

• Manríquez, J., Nguyen, N.-C., Solano, M., A dissimilar non-matching HDG discretization for Stokes flows, Comput. Methods Appl. Mech. Engrg, 399 (2022), Article 115292, https://doi.org/10.1016/j.cma.2022.115292.
• Bermúdez, I., Manríquez, J., Solano, M., An HDG method for Stokes-Darcy coupling in dissimilar meshes, manuscript in preparation (2024).