Seminar
Splitting methods for coupled problems based on Robin-Robin coupling conditions
We develop a loosely coupled splitting method for coupled problems called the Robin-Robin splitting method based on Robin-type interface conditions. We apply this method to several coupled problems, most notably the fluid-structure interaction problem in both a time-semi-discrete framework and a fully discrete framework.
Pitt Career Center Talk
If you are considering your post-graduation options or are curious as to what opportunities Pitt can provide you to prepare for your future career, this talk is for you. Lauren will discuss how to connect with the Career Center and the services it offers. Note also that this talk comes ahead of the career fair, which will be held on September 28th-30th.
All students are welcome to attend and there will be pizza.
Multiphase Mean Curvature Flow: Uniqueness Properties of Weak Solution Concepts and Phase-Field Approximations
Topology changes occur naturally in geometric evolution equations like mean curvature flow. As classical solution concepts break down at such geometric singularities, the use of weak solution concepts becomes necessary in order to describe topological changes.
For two-phase mean curvature flow, the theory of viscosity solutions by Chen-Giga-Goto and Evans-Spruck provides a concept of weak solutions with basically optimal existence and uniqueness properties.
Jerome Wettstein - Half-Harmonic Gradient Flows: Existence, Uniqueness and Bubbling
Organizational Meeting
Approximation of fractional operators and fractional PDEs using a sinc-basis
We introduce a spectral method to approximate PDEs involving the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of O(NlogN) for the application of the operator to a discretization with N degrees of freedom.
Constructing robust high order entropy stable discontinuous Galerkin methods
High order methods are known to be unstable when applied to nonlinear conservation laws whose solutions exhibit shocks and turbulence. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of numerical resolution or solution regularization and shock capturing.