## GEOMETRIC PDEs AND THEIR APPROXIMATIONS

### January 10-16, Department of Mathematics, Texas A&M

### Course Description

This winter class consists of plenary lectures and computer laboratory sessions.

Deformable domains are ubiquitous in several areas of research but are still a serious computational challenge. The morning lectures provide a comprehensive approach to the formulation and discretization of geometric partial differential equation governing processes. The topics discussed center around differential geometry, calculus of variations, gradient flows and discretization using finite element methods. Emphasis is made to application and in particular to biophysics and material science.

The laboratory sessions serve as numerical illustration and provide insights on the behavior of the algorithms introduced in class. During these sessions students will have the choice of using either MATLAB or deal.ii/C++. A deal.ii tutorial will be provided on Sunday January 10, 2016.

### Lecturers

The morning lectures are given by Ricardo H. Nochetto (University of Maryland) and Andrea Bonito (Texas A&M). The deal.ii tutorial on January 10 is given by Timo Heister (Clemson University). The lab sessions are covered by S. Patty (TAMU) and W. Lei (TAMU).

### Participants

A list of confirmed participants can be found here.

### Fees

Participation is free of charge.

### Support - DEADLINE October 31st, 2015

**Not available anymore**. *Some limited funding is available to cover all expenses including travel up to USD500 for US participants and up to USD700 for non US participants. Application should be made during the registration process and requires the candidate CV as well as the email address of one reference. Priority will be given to graduate students. Applications from women, minorities, individuals with disabilities, and veterans are strongly encouraged.*

### Tentative Schedule

Sun, Jan 10 | Mon, Jan 11 | Tue, Jan 12 | Wed, Jan 13 | Thu, Jan 14 | Fri, Jan 15 | |
---|---|---|---|---|---|---|

09:00 -- 10:30 | Registration (10:00) | Lecture 1 | Lecture 4 | Lecture 6 | Lecture 8 | Lecture 10 |

10:30 -- 10:45 | Registration | Break | Break | Break | Break | Break |

10:45 -- 12:45 | deal.ii Tutorial 1 | Lecture 2 | Lecture 5 | Lecture 7 | Lecture 9 | Lecture 11 |

12:45 -- 13:30 | Lunch | Lunch | Lunch | Lunch | Lunch | Lunch |

13:30 -- 15:00 | deal.ii Tutorial 2 | Lecture 3 | Lab Session 2 | Lab Session 4 | Lab Session 6 | Lecture 12 |

15:00 -- 15:15 | Break | Break | Break | Break | Break | Break |

15:15 -- 16:45 | deal.ii Tutorial 3 | Lab Session 1 | Lab Session 3 | Lab Session 5 | Lab Session 7 | Lab Session 8 |

16:45 -- 17:00 | -- | Break | Break | Break | Break | -- |

17:00 -- 18:00 | -- | Fac Q/A | Fac Q/A | Fac Q/A | Fac Q/A | -- |

##### Lectures (Bloc 166 in Blocker building)

Lectures 1,2&3: | Introduction and Differential Geometry I |

Lectures 4&5: | Differential Geometry II |

Lectures 6&7: | Finite Element Methods for the Laplace-Beltrami Operator |

Lectures 8&9: | Shape Differential Calculus |

Lectures 10&11: | Application in Math-Biology - Biomembranes |

Lecture 12: | Application in Material Science - Bilayer Plates Bending Actuators |

##### Lab Sessions (Bloc 128 and 133)

The participants chose between matlab or deal.ii projects. In both cases, the Lab Session I is devoted to a review of the finite element method and its implementation.

- The deal.ii project (bloc 133) builds on the knowledge aquired during the tutorials and consists in the implementation of adaptive higher order finite element methods for the Laplace-Beltrami operator in three dimension.
- The matlab project (bloc 128) do not require any specific knowledge and consists in three different linear finite element based discretization methods for the Laplace-Beltrami operator: parametric method, first order narrow band level-set method and phase field method. Code snipsets are provided in each case to serve as starting points.

##### Facultative Questions / Answers (Fac Q/A)

Additional materials and exercises are given during the lectures. The faculatitive questions and answers time slots can be used to ask questions to the lecturer about them.

### Accommodation Arrangements

The official winter school hotel is the Home2Suite. Alternatively, we recommend the Hampton Inn or the Super 8 for smaller budgets. In all cases, make sure to ask whether state rate rooms are available.

### Acknowledgments

The winter school is partially supported by the National Science Foundation, the Institute for Applied Mathematics and Computational Science, the College of Science and the department of mathematics at Texas A&M.

### Contact

Please email any question/comment/concern to Andrea Bonito