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Exact domain truncation for scattering problems
Abstract
While scattering problems are posed on unbounded domains, volumetric discretizations typically require truncating the domain at a finite distance, closing the system with some sort of boundary condition. These conditions typically suffer from some deficiency, such as perturbing the boundary value problem to be solved or changing the character of the operator so that the discrete system is difficult to solve with iterative methods.
We introduce a new technique for the Helmholtz problem, based on using the Green formula representation of the solution at the artificial boundary. Finite element discretization of the resulting system gives optimal convergence estimates. The resulting algebraic system can be solved effectively with a matrix-free GMRES implementation, preconditioned with the local part of the operator. Extensions to the Morse-Ingard problem, a coupled system of pressure/temperature equations arising in modeling trace gas sensors, will also be given.
Intrinsic models on Riemannian manifolds with bounded curvature
Abstract
We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modeled by an interaction potential. Without the diffusion term, we consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution’s support. With the diffusion term, the attractive interaction and the diffusion compete. We provide the conditions of the attractive interaction for each part to win.
On the Incompressible Limit for a Tumour Growth Model Incorporating Convective Effects
Abstract
In this seminar, we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporated the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model, which builds a bridge between the density-based model and a geometry free-boundary problem by passing to a singular limit in the pressure law. The limiting objects are then proven to be unique.