Wednesday, 23 April, 2014
SPEAKER: Mr. Cody Lorton, UT
TITLE: An Efficient Numerical Method for Wave Scattering in Random Media
ABSTRACT: Wave scattering in random media arises in many scientific and engineering field including geoscience, materials science and medical science. Computing quantities of interest for the solutions of such wave problems, especially, in the high frequency case, poses a daunting computational challenge because of sheer amount of computations required to solve those problems. Due to their strong indefiniteness, highly oscillatory nature of solutions, and lack of efficient iterative solvers, standard numerical approaches such as brute force Monte Carlo methods and stochastic Galerkin methods are either too expensive to use or do not work well. In this talk we shall present a newly developed multi-resolution approach for the random Helmholtz problem with large wave numbers. In this approach the original random Helmholtz problem is reduced to a finite number of nearly deterministic and non-homogeneous Helmholtz problems with random source terms, which are discretized by some unconditionally stable discontinuous Galerkin methods. An efficient solver with computational complexity of order O(3N^3/2) is also proposed to solve the resulting algebraic problems. Convergence analysis and numerical experiments will be presented to demonstrate the potential advantages of the proposed numerical approach.
SPEAKER: Mr. Yukun Li, UT
TITLE: Discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow
ABSTRACT: This talk is concerned with some new convergence results for interior penalty discontinuous Galerkin (IPDG) approximations of the Allen-Cahn and its sharp interface limit known as the mean curvature flow. The main result to be presented is the convergence of the numerical interfaces to the sharp interface of the mean curvature flow as both the numerical mesh parameters and the phase field parameter (called the interaction length) tend to zero. The crux for establishing this result is to derive, by a nonstandard technique, error estimates for the IPDG solutions which blows up only polynomially (instead of exponentially) in the reciprocal of the phase field parameter. Numerical experiments will also be presented to gauge the performance of the proposed IPDG methods.