Numerical Analysis Seminar

Date: October 31, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Juan-Pablo Borthagaray, University of Maryland

Title: Finite elements for fractional diffusion: towards nonlinear problems

Abstract: In this talk we consider problems involving the integral fractional Laplacian on bounded domains. The first part is devoted to analysis of linear problems; we discuss regularity of solutions, analyze direct finite element implementations and derive convergence rates. Afterwards we discuss two nonlinear problems: the fractional obstacle problem and the computation of nonlocal minimal surfaces. The integral fractional Laplacian is a nonlocal operator given by a singular integral (defined in the principal value sense). Therefore, suitable quadrature is required to handle the singularity of the kernel. Nonlocality originates additional difficulties, such as the need to cope with integration on unbounded domains and full stiffness matrices. Independently of the smoothness of the domain and the data, solutions to the problems under consideration possess a limited Sobolev regularity. In order to enhance the order of convergence of the finite element approximations, we introduce suitably defined weighted Sobolev spaces. This, in turn, leads to the consideration of discrete solutions on graded meshes and permits to obtain optimal convergence rates in two-dimensional domains.