Wednesday,
November 9, 12:00 p.m. GWC 487
Jeff Springer
Northern Arizona University
Solutions of Semilinear Elliptic PDEs on Manifolds
Abstract
In this talk we introduce the Closest Point Method (CPM) and
the Gradient Newton Galerkin Algorithm (GNGA). The Closest Point Method
(CPM) is a recent embedding method for solving time-dependent PDE’s on
surfaces, which can also easily be utilized to solve eigenvalue problems on surfaces.
Implementation of CPM to solve problems on surfaces simplifies possible
geometric complications as no prior knowledge of the surface is required, other
than it’s closest points. The Gradient Newton Galerkin Algorithm (GNGA)
is a method for finding approximations of critical points of a functional. This
method was introduced by Neuberger and Swift to solve PDEs of the form
Δu + f(u) = 0 on regions Ω with zero Dirichlet boundary conditions. We use
GNGA to solve PDEs on manifolds by first finding the eigenfunctions of the
Laplacian restricted to the manifold using the CPM. Finally we use a Galerkin
expansion, in eigenfunctions of the Laplacian, to find the desired solutions. Examples
on basic manifolds will be presented along with suggestions for future
work.
