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Archive : Numerical Analysis and PDE

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Harbir Antil, George Mason UniversityHarbir Antil, George Mason UniversityEWG 336Title: Fractional linear and semilinear PDEs: analysis, control and discretization <br></br> Abstract: Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. In this talk we will provide an introduction to fractional derivatives. We will discuss how to incorporate nonzero boundary conditions in fractional PDEs and study the boundary optimal control problems. We will conclude this talk with analysis and error estimates for fractional semilinear problems.12/7/2017 4:00:00 PM12/7/2017 5:00:00 PMFalse
Juan Pablo Borthagaray, University of Maryland College ParkJuan Pablo Borthagaray, University of Maryland College ParkEWG 336Title: Finite element approximations of the nonhomogeneous fractional Dirichlet problem <br></br> Abstract: We study finite element approximations of the following non-homogeneous Dirichlet problem for the integral fractional Laplacian on a bounded domain in n-dimensional space. We analyze two different approaches: the first one consists on the direct imposition of the Dirichlet condition; the second one is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both for the solution and its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter. This is joint work with Gabriel Acosta (UBA, Argentina) and Norbert Heuer (PUC, Chile).11/30/2017 4:00:00 PM11/30/2017 5:00:00 PMFalse
Charles Epstein, University of Pennsylvania Charles Epstein, University of Pennsylvania Ewing 336Title. The stability of time-domain integral equations for acoustic wave propagation <br></br> Abstract. We give a principled approach for the selection of a boundary integral, retarded potential representation for the solution of scattering problems for the wave equation in an exterior domain. This is joint work with Leslie Greengard and Tom Hagstrom 11/16/2017 4:30:00 PM11/16/2017 5:30:00 PMFalse
Lise-Marie Imbert Gerard, New York University and University of Maryland College Park Lise-Marie Imbert Gerard, New York University and University of Maryland College Park Ewing 336Title: Pseudo-spectral methods on surfaces of genus one Abstract: We will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.11/9/2017 4:00:00 PM11/9/2017 5:00:00 PMFalse

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