#### Date of Award

Fall 2013

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Jie Shen

#### Committee Chair

Jie Shen

#### Committee Member 1

Patricia Bauman

#### Committee Member 2

Peijun Li

#### Committee Member 3

Robert Skeel

#### Abstract

This dissertation focuses on the development of high-order numerical methods for acoustic and electromagnetic scattering problems, and nonlinear fluid-structure interaction problems.

For the scattering problems, two cases are considered: 1) the scattering from a doubly layered periodic structure; and 2) the scattering from doubly layered, unbounded rough surface. For both cases, we first apply the transformed field expansion (TFE) method to reduce the two-dimensional Helmholtz equation with complex scattering surface into a successive sequence of the transmission problems with a plane interface. Then, we use Fourier-Spectral method in the periodic structure problem and Hermite-Spectral method in the unbounded rough surface problem to reduce the two-dimensional problems into a sequence of one-dimensional problems, which can then be efficiently solved by a Legendre-Galerkin method.

In order for TFE method to work well, the scattering surface has to be a sufficiently small and smooth deformation of a plane surface. To deal with scattering problems from a non-smooth surface, we also develop a high-order spectral-element method which is more robust than the TFE method, but is computationally more expensive.

We also consider the non-linear fluid-structure interaction problem, and develop a class of monolithic pressure-correction schemes, based on the standard pressure-correction and rotational pressure-correction schemes. The main advantage of these schemes is that they only require solving a pressure Poisson equation and a linear coupled elliptic equation at each time step. Hence, they are computationally very efficient. Furthermore, we prove that the proposed schemes are unconditionally stable.

#### Recommended Citation

He, Ying, "Efficient Spectral-Element Methods For Acoustic Scattering And Related Problems" (2013). *Open Access Dissertations*. 148.

http://docs.lib.purdue.edu/open_access_dissertations/148