Applied Mathematics Colloquium: Dr. Thi Thao Phuong Hoang

Title: Domain Decomposition and Local Time-Stepping Methods For Flow and Transport in Porous Media

Domain decomposition (DD) methods provide a natural computational framework for multiscale multiphysics problems and a powerful tool for numerical simulation of large-scale problems. As many physical and engineering processes are described by evolution partial differential equations, extensions of DD methods to dynamic systems (i.e. those changing with time) have been a subject of great interest. Moreover, for applications in which the time scales vary considerably across the whole domain due to changes in the physical properties or in the spatial grid sizes, it is critical and computationally efficient to design DD methods which allow the use of different time step sizes in different subdomains. 

In this talk, we will introduce mathematical concepts of DD methods for evolution equations, and present global-in-time, nonoverlapping domain decomposition methods for the linear advection-diffusion equation (with operator splitting) and for the nonlinear Stokes-Darcy system. Two types of methods are studied: one is based on a generalization of the Steklov-Poincaré operator to time-dependent problems and the other is based on the Optimized Schwarz Waveform Relaxation (OSWR) method in which more general (Robin or Ventcell) transmission conditions are used to accelerate the convergence of the method. Both mathematical analysis and numerical performance of these methods with nonconforming time grids will be investigated.