These lectures will be given by Professor Barbara Wohlmuth from the Technical University of Munich (TUM). The aim of the lectures is to give an insight into the mathematical and numerical challenges of non-linear or non-local partial differential equations in applications. We use a variety of applications ranging from porous-media flow systems, fluid and structural mechanics to finance. The special challenges of variational inequalities, mixed-dimensions and non-integer differential operators are addressed. We illustrate the flexibility of abstract mathematical concepts and discuss limitations in theory and convergence.
Key questions that will be addressed are:
- What are the abstract variationally consistent formulation?
- How to couple the different dimensions?
- How to deal with the long term history and the memory effect?
Lecture I: Variational inequalities
- Examples for surface and volume constraints
- From inequality constraints to equality
- A priori convergence results
- The Newton convergence
Lecture II: Mixed-dimensional systems
- Examples in applications
- The coupling concept
- The challenge of spaces
- Second gradient terms
Lecture III: Non-local operators
- Random materials as example
- Random wind generation as example
- The effect of subdiffusion
- The numerical challenge