Plenary Lecture 2

Beatrice Rivière (Rice University)

Title:

Multi-dimensional Coupled PDEs: Theory and Application

Abstract:

Multi-Dimensional coupled problems are characterized by coupled partial differential equations defined over domains of different dimensions. These problems  occur in several applications ranging from geosciences to biomedicine. Drug delivery from the vasculature to the organ, or solute clearance through the lymphatic vessels are two such examples of processes involving flow and transport in networks of one-dimensional vessels embedded in a three-dimensional domain. 

This talk presents recent advances for the numerical analysis of multi-dimensional PDEs with co-dimension equal to two. We first discuss the case of elliptic partial differential equations with line source. The analysis of such problems is non-standard because the solution exhibits a logarithmic singularity near the 1D line. Convergence of a discontinuous Galerkin scheme is obtained. Second, we present the convergence of a discrete scheme for the fully coupled 3D-1D elliptic equations.  The analysis is based on deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators.  Finally the numerical methods are applied to model flow in liver.