Time-Harmonic Maxwell's equations
Speaker: Kshitij Sinha, IIT Bombay
Host: Harsha Hutridurga
Title: Time-Harmonic Maxwell's equations - Regularity theory and related topics
Time, day and date: 11:00:00 AM – 12:00:00 PM, Monday, January 12
Venue: Ramanujan Hall
Abstract: In the first seminar of this series, we propose to introduce the time-harmonic Maxwell's equations. Few necessary ingredients to the rigorous study of these equations such as the Friedrich's inequality, the Helmholtz decomposition for L2 vector fields will be discussed. In this talk, we also plan to prove a conditional well-posedness results for these equations.

Stefan Problem Seminar
Speaker: Neeraj Rawat, IIT Bombay
Host: Harsha Hutridurga
Title: Stefan problem - A free boundary value problem
Time, day and date: 11:00:00 AM – 12:30:00 PM, Tuesday, January 13
Venue: Ramanujan Hall
Abstract: In the first Lecture of this series, we will prove the existence and uniqueness of classical solution to the one-phase stefan problem in dimension one.Throughout this series of lectures, we will mainly follow the work of Avner Friedmann.

Short Course
Speaker: Prof. Jerome Droniou, University of Montpellier
Host: Parthanil Roy
Title: Introduction to a polytopal complex: the Discrete De Rham method
Time, day and date: 9:30:00 AM – 11:00:00 AM, Thursday, January 15
Venue: Ramanujan Hall (https://us06web.zoom.us/j/6950684207?pwd=VkNmYzFBd0c0QWx3azhJODJ0QUp1Zz09&omn=85848985788 ID: 695 068 4207 Passcode: 802747)
Abstract: I will start by explaining what the de Rham complex is and, through examples, why it is important for the theoretical analysis of certain partial differential equations involving gradient, curl or divergence operators.
I will then present the Discrete De Rham (DDR) method, an arbitrary-order discretisation of the de Rham complex that is applicable on generic polyhedral meshes. We will see that the construction of the method is entirely based on mimicking integration-by-parts properties (for the gradient, the curl and the divergence), and is hierarchical: we construct the space and discrete operators starting from the edges, then going on the faces, then on the elements.
I will highlight the main properties of the DDR method, which allows for convergence analysis of schemes built on it. A particular focus will be on the notion of "adjoint consistency" which arises due to the non-conformity of the method. I will detail such a convergence analysis on the example of the Stokes equations in curl-curl formulation.

Coagulation-Fragmentation equations
Speaker: Ram Gopal Jaiswal, IIT Bombay
Host: Harsha Hutridurga
Title: A first course on the Coagulation-Fragmentation equations.
Time, day and date: 11:00:00 AM – 12:30:00 PM, Thursday, January 15
Venue: Ramanujan Hall
Abstract: This series of lectures focuses on the existence and uniqueness of weak solutions to the continuous coagulation equation under suitable assumptions on the coagulation kernel. We study separately the regimes in which solutions conserve total mass and those in which mass conservation may fail in finite time due to a phenomenon known as gelation. We then incorporate fragmentation processes, treating linear and nonlinear (collision-induced) fragmentation separately, and establish the corresponding existence and uniqueness results. Moreover, we prove the existence of mass-conserving stationary (equilibrium) solutions to coagulation equations with linear fragmentation under appropriate assumptions on the coagulation and fragmentation kernels.

Short Course
Speaker: Prof. Jerome Droniou, University of Montpellier
Host: Parthanil Roy
Title: Introduction to a polytopal complex: the Discrete De Rham method
Time, day and date: 4:00:00PAM – 05:30:00PAM, Thursday, January 15
Venue: Ramanujan Hall (https://us06web.zoom.us/j/6950684207?pwd=VkNmYzFBd0c0QWx3azhJODJ0QUp1Zz09&omn=85848985788 ID: 695 068 4207 Passcode: 802747)
Abstract: I will start by explaining what the de Rham complex is and, through examples, why it is important for the theoretical analysis of certain partial differential equations involving gradient, curl or divergence operators.
I will then present the Discrete De Rham (DDR) method, an arbitrary-order discretisation of the de Rham complex that is applicable on generic polyhedral meshes. We will see that the construction of the method is entirely based on mimicking integration-by-parts properties (for the gradient, the curl and the divergence), and is hierarchical: we construct the space and discrete operators starting from the edges, then going on the faces, then on the elements.
I will highlight the main properties of the DDR method, which allows for convergence analysis of schemes built on it. A particular focus will be on the notion of "adjoint consistency" which arises due to the non-conformity of the method. I will detail such a convergence analysis on the example of the Stokes equations in curl-curl formulation.