All problems described below have been tested with the VoronoiFVM.jl package for the solution of systems of partial differential equations which will be introduced during the course.
Alternatively you can solve these using the Julia finite element package Gridap.jl which I hopefully will find time to partially cover in the lecture. I that case, I would need to check if this can work before agreeing with this.
During a project, you should implement the particular problem, provide examples of numerical solutions and present a written report. A pdf generated from a Pluto notebook will be accepted. You shall present the details of the underlying space and time discretization approaches and the strategy for the solution of the linear systems of equations. All work is ideally performed in groups of 3 students, who then take can the exam together.
Chemotaxis (nonlinear, PDE system):
Chemotaxis is linked to signalling and movement of bacteria. It is assumed that bacteria with a concentration u send out some chemicals with concentration which attract other bacteria of the same kind. Work will be based on the minimal model M1 in this paper
Bidomain problem (nonlinear, PDE system):
This system of equations describes the transport electrical excitations through the tissue of the heart. Work will be based on this paper.
Heterogeneous catalysis (nonlinear, PDE system):
Catalysis is a process where a chemical reaction which for given conditions (pressure, temperature) would not occur due to the energy barrier needed to initiate it is facilitated by a catalyst. Heterogeneous catalysis means that this process happens at the boundary of the domain (e.g. a platinum surface) Work will be based on a system of two diffusion equations coupled by a catalytic reaction at the boundary, a simplified version of the eqautions in this paper.
Solute transport (nonlinear, PDE system)
Solute transport in a saturated porous medium (ignoring gravity), described by a system of two equations: Darcy's law for the flow of water and a coupled convection-diffsion problem for the transport of dissolved material.
Groundwater flow (nonlinear)
Richards equation of saturated/unsaturated flow in a porous medium, described by a Darcy's law with a nonlinear diffusion coefficient.
Porous medium heated from below (nonlinear system)
Coupled system of Darcy's law with variable density and and heat transport
Parameter identification (can be combined with different problems)
Identify parameters (e.g. reaction constants) e.g. in a reaction-diffusion system based on given mock measurements
Poisson-Boltzmann problem (nonlinear)
A nonlinear Poisson equation describing equilibrium charge distribution in e.g. a semiconductor
Porous medium equation (nonlinear)
A nonlinear diffusion equation describing gas flow in a porous medium
Bring your own problem you are interested in (Note: free (non-porous medium) flow problems are lesss easy to handle here). Please give me a short description of the idea, so that I can judge the feasibility.
Assessment of linear solver performance (linear)
Perform convergence rate experiments for linear (direct and iterative) solvers for discretized linear PDEs in 1/2/3D
Comparison of time discretization strategies.
In the lecture, I will introduce several simple time discretization strategies and focus on the implicit Euler method. VoronoiFVM.jl can be coupled with the DifferentialEquations.jl ecosystem which provides a vast number of time discretization methods. Especially problems 1,2,6 and 9 can take advantage of these.