Doctoral Dissertations

Date of Award

5-2001

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Xiaobing Feng

Committee Members

Ohannes Karakashian, Henry Simpson, Chris Pionke

Abstract

The following thesis studies the acoustic wave equation, the elastic wave equa-tions, a fluid-solid interaction problem, and their finite element approximations in the frequency domain. The focus is on how the solutions depend on the frequency uj, how the error bounds for the finite element approximations depend on the frequency u, and how the mesh size h is constrained by the frequency ω in the finite element approximations. Particular emphasis is on results for high frequency waves. A Rellich identity technique is used to derive an elliptic regularity estimate for the acoustic Helmholtz equation with a first order absorbing boundary condition. The estimate is optimal with respect to the frequency ω. The finite element method for the problem is formulated and analyzed. The finite element analysis leads to two main results. The first is a constraint on the mesh size h in terms of the frequency ω which is necessary to guarantee existence of finite element approximations. The second is an error bound on the finite element approximations which shows explicit ω dependence. Analogous techniques achieve similar results for the elastic Helmholtz equations. An additional difficulty appears in the elastic case because the Lamé operator is only semi-positive definite. The difficulty is overcome first with a regularity argument, and the result is then improved with a Korn-type inequality on the boundary. A fluid-solid interaction problem, which is described by a coupled system of acous-tic and elastic Helmholtz equations, is considered next. Finite element approxima-tions are proposed and analyzed, and optimal order error estimates are established. Parallelizable iterative algorithms are proposed for solving the corresponding finite element equations. The algorithms are based on domain decomposition methods. Strong convergence in the energy norm of the algorithms is proved. Finally, the acoustic Helmholtz equation with a second order absorbing boundary condition is studied. Again, the finite element method is formulated and analyzed, and optimal error estimates are derived with explicit dependence on the frequency, ω. A procedure for recovering the solution in the time domain by numerically approx-imating the inverse Fourier transform is formulated. The procedure is implemented for both the acoustic Helmholtz problem with the first order absorbing boundary condition, and for the acoustic Helmholtz problem with a second order absorbing boundary condition. A computational comparison of the resulting approximate solu-tions is given.

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