Analysis of finite element based numerical methods for acoustic waves, elastic waves, and fluid-solid interactions in the frequency domain.

摘要

We study the acoustic wave equation, the elastic wave equations, 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 o, how the error bounds for the finite element approximations depend on the frequency o, and how the mesh size h is constrained by the frequency o in the finite element approximations. 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 o. The finite element method for the problem is formulated and analyzed. Finite element analysis leads to two main results. The first is a constraint on the mesh size h in terms of the frequency o, which is necessary to guarantee existence of finite element approximations. The second is an error bound which shows explicit o dependence.;Analogous techniques achieve similar results for the elastic Helmholtz equations. An additional difficulty appears in the elastic case because the Lame operator is only semi-positive definite. The difficulty is overcome with a regularity argument, and the result is improved with a Korn-type inequality.;A fluid-solid interaction problem, which is described by a coupled system of acoustic and elastic Helmholtz equations, is considered next. Finite element approximations 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, o. A procedure for recovering the solution in the time domain by numerically approximating the inverse Fourier transform is formulated. The procedure is implemented with both the first and second order absorbing boundary condition. A computational comparison of the resulting approximate solutions is given.