Challenges in Geometry , Analysis and Computation : High Dimensional Synthesis

We present a new technique, based on a clever numerical minimization of certain Sobolev norms, for the numerical solution of partial differential equations. The new method is very high order even for complex geometries and variable coefficient problems. One surprising feature of the method is that we prefer to pose the problem in first-order form. The system does not have to be square as long as the equations are linearly independent and possess a unique solution. We show the efficiency of the method for computing highly accurate solutions by applying it to various classical 2D PDEs including: div-curl, Poisson (both 4x3 and 3x3 formulation), Helmholtz, wave equation (1+1), heat equation (1+1), elastic equation, biharmonic, and linearized Navier-Stokes. What is more amazing about this method is that we use the same code for all of these problems, unlike classical FEM and FD codes that have to be re-written to match the problem type. The entire code is only a few hundred lines of Matlab. We will also solve a div-curl problem with a singular solution that is simultaneously discontinuous on a slit. The problem will be solved with both a single boundary condition and a double boundary condition on the slit. In all cases we will retain high order convergence and show very high accuracy in the solution (including biharmonic problems). The geometries we consider will be very complex, including the exterior region of a car shaped body. We will also show high order convergence on curved geometries. Presenter: Yen Do (Yale University) Title: Quantitative convergence of Fourier series in weighted settings. Abstract: I will describe some ongoing joint work with Michael Lacey about variational Carleson theorems in weighted spaces. I will describe some ongoing joint work with Michael Lacey about variational Carleson theorems in weighted spaces. Presenter: Adrianna Gillman (Darmouth College) Title: A fast direct solution technique for two-dimensional quasi-periodic fields. Abstract: Accurate numerical modeling of periodic scattering problems is important for many applications including solar cells, diffraction gratings, acoustics absorbers, and photonic crystal slabs. Recently, a new integral representation for quasi-periodic scattering problems that achieves spectral accuracy has been developed. This representation avoids the blow-up of the quasi-periodic Greens function at so-called Wood anomalies by restricting the exterior domain to a quasi-periodic Accurate numerical modeling of periodic scattering problems is important for many applications including solar cells, diffraction gratings, acoustics absorbers, and photonic crystal slabs. Recently, a new integral representation for quasi-periodic scattering problems that achieves spectral accuracy has been developed. This representation avoids the blow-up of the quasi-periodic Greens function at so-called Wood anomalies by restricting the exterior domain to a quasi-periodic