Solving partial differential equations (PDEs) is crucially important in many disciplines within science, engineering and even in economics. Next to these disciplines, more and more biological processes are simulated using PDE-based frameworks. Examples are contraction of skin that develops after a serious burn injury, or the simulation of pattern formation of the skin of zebrafish or tigers. Even on the smallest units of life, e.g. cells, PDEs are solved to simulate processes like cell migration, and the impact that cells inflict on their immediate environment. This impact may be caused by chemicals that are released by cells, or by forces that are exerted on their environment during migration. PDEs often occur as conservation laws for energy, momentum or mass in body-like structures. Next to solving PDEs in bulks, PDEs may also govern on very thin shells or layers inside the bulk or surrounding the bulk. If the thickness of the layer is much smaller than the dimensions of the bulk, then one typically neglects the thickness and treats the layer as a surface within the bulk or around the bulk, that is, the surface has zero measure in the body. In various applications, this layer may separate two different regimes or phases. In these cases, the layer is commonly referred to as an interface, and this interface may move within the domain of computation over time. Various numerical approaches exist to deal with such moving interfaces. In this summer school, we treat the phase field method as one of the most-widely used methods to deal with moving interfaces. In many applications, the (interfacial) surface may, furthermore, be subject to diffusional transport over the surface, which results into a PDE over the surface. The resulting Surface PDEs (SPDEs) are composed by gradient and divergence operators, as well as Laplace-Beltrami operators over the surface. The numerical solutions to SPDEs can help predicting relevant results that are difficult or impossible to forecast on the basis of experiments. Often the solutions over the surface PDE is directly coupled to the solution of bulk PDE.
In this summer school, we focus on numerical (predominantly finite element) strategies applied to (bulk-)surface PDEs and phase-field problems. The finite element method is based on variational principles and allows a large degree of geometrical flexibility and flexibility regarding possible jumps of material constants. The summer school will focus on implementation and development of bulk-surface (finite element) discretizations. The goal of this summer school is to give the participants an introduction to the (bulk-)surface finite element and other discretization methods for phase field problems. The course will be centered around the following elements:
- Understanding the core definition of the polynomial (Lagrangian) finite element spaces;
- Understanding the gradient, divergence and Laplace-Beltrami differential operators on manifolds;
- Understanding and implementing a (bulk-)surface method for a proto-problem;
- Understanding and implementing finite difference methods for phase-field problems;
- Some aspects from today’s state-of-the-art research in (moving) (bulk-)surface problems.
The summer school will therefore be amenable to (applied) mathematicians, engineers and scientists. The summer school will be open to participants having a basic knowledge of calculus and partial differential equations as they are taught to engineers and scientists. All sessions will be hands-on, meaning that sufficient time for exercises and their discussion is planned. Lecturers will be experts in the field: Anotida Madzvamuse (University of British Columbia, Canada), Chandrasekhar Venkataraman (University of Sussex, UK), Davide Cuseddu (University of Lisbon, Portugal), Massimo Fritelli (University of Salento, Italy) and Sebastian Aland (University of Freiberg, Germany).
The duration of the summer school is four days. Each day will be divided in a theoretical session in the morning, which is followed by hands-on assignments in the afternoon.
Lecture: Basic notions of finite element methods (weak form, Galerkin method, Basis functions), introduction into surface PDEs
Lab work: Matlab session to write bulk finite element code in 2D (3D)
Lecture: Continuation of surface PDEs, Basic notions of surface finite element methods (weak form), method of lines (time integration)
Lab work: Matlab session to write surface finite element code in 2D (3D)
Lecture: Bulk-surface finite elements, treatment of moving surface element methods
Lab work: Fenics project for time-dependent bulk-surface finite elements (application) in 2D (3D)
Lecture: Introduction to Phase-field methods for moving interface problems
Lab work: finite differences for phase-field methods