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

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

On the robust exponential convergence of hp finite element methods for problems with boundary layers

The hp version of the finite element method for a one-dimensional, singularly perturbed elliptic-elliptic model problem with analytic input data is considered. It is shown that the use of piecewise polynomials of degree p on a mesh consisting of three suitably chosen elements leads to robust exponential convergence, i.e., the exponential rate of convergence depends only on the input data and is independent of the perturbation paramete

Finite element schemes for elliptic boundary value problems with rough coefficients

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the task of computing reliable numerical approximations of the solutions of elliptic equations and systems where the coefficients vary discontinuously, rapidly, and by large orders of magnitude. Such problems, which occur in diffusion and in linear elastic deformation of composite materials, have solutions with low regularity with the result that reliable numerical approximations can be found only in approximating spaces, invariably with high dimension, that can accurately represent the large and rapid changes occurring in the solution. The use of the Galerkin approach with such high dimensional approximating spaces often leads to very large scale discrete problems which at best can only be solved using efficient solvers. However, even then, their scale is sometimes so large that the Galerkin approach becomes impractical and alternative methods of approximation must be sought. In this thesis we adopt two approaches. We propose a new asymptotic method of approximation for problems of diffusion in materials with periodic structure. This approach uses Fourier series expansions and enables one to perform all computations on a periodic cell; this overcomes the difficulty caused by the rapid variation of the coefficients. In the one dimensional case we have constructed problems with discontinuous coefficients and computed the analytical expressions for their solutions and the proposed asymptotic approximations. The rates at which the given asymptotic approximations converge, as the period of the material decreases, are obtained through extensive computational tests which show that these rates are fundamentally dependent on the level of regularity of the right hand sides of the equations. In the two dimensional case we show how one can use the Galerkin method to approximate the solutions of the problems associated with the periodic cell. We construct problems with discontinuous coefficients and perform extensive computational tests which show that the asymptotic properties of the approximations are identical to those observed in the one dimensional case. However, the computational results show that the application of the Galerkin method of approximation introduces a discretization error which can obscure the precise asymptotic rate of convergence for low regularity right hand sides. For problems of two dimensional linear elasticity we are forced to consider an alternative approach. We use domain decomposition techniques that interface the subdomains with conjugate gradient methods and obtain algorithms which can be efficiently implemented on computers with parallel architectures. We construct the balancing preconditioner, M,, and show that it has the optimal conditioning property k(Mh(^-1)Sh) = 0 is a constant which is independent of the magnitude of the material discontinuities, H is the maximum subdomain diameter, and h is the maximum finite element diameter. These properties of the preconditioning operator Mh allow one to use the computational power of a parallel computer to overcome the difficulties caused by the changing form of the solution of the problem. We have implemented this approach for a variety of problems of planar linear elasticity and, using different domain decompositions, approximating spaces, and materials, find that the algorithm is robust and scales with the dimension of the approximating space and the number of subdomains according to the condition number bound above and is unaffected by material discontinuities. In this we have proposed and implemented new inner product expressions which we use to modify the bilinear forms associated with problems over subdomains that have pure traction boundary conditions.This work is funded by the Engineering and Physical Sciences Research Council

Design, stability and applications of two dimensional recursive digital filters

Imperial Users onl

Two Level Additive Schwarz Preconditioner For Control Volume Finite Element Methods

In this thesis we investigate nummerically the convergence properties of the control volume finite element method (CVFEM) preconditioned with a two level overlapping additive Schwarz method. Relevant theory regarding the CVFEM, the Schwarz framwork and the iterative solver Genral Minimal Residual Method is explained

Modeling a fish population with diffusive and advective movement in a spatial environment

This dissertation has developed an individual-based, physiologically structured model for a fish population with diffusive and advective movement in a spatial environment. It incorporates spatio-temporal processes and individual processes simultaneously into the population dynamic model of a McKendrick-von Foerster type partial differential equation. Anindividualfish is physiologically structured according to age, lipid and structure (protein and carbohydrates). Fish are assumed to be immobile in their embryonic stage and the fish begin to feed and might move after the embryonic stage. Advective processes are induced by environmental heterogeneity, in which fish move toward neighboring areas with different levels of, for instance, resource density or/and chemical toxicant concentration. The population dynamic model is complicated, in that it is a mixed type partial differential equation that combines a quasi-linear hyperbolic equation in the embryonic stage and degenerate parabolic equation in the older life stage. Some mathematical aspects of the model of primary interest have been discussed. The existence of a local weak solution has been shown. By the constructive analysis used to demonstrate the existence of a local solution, a computational scheme for the mathematical model has been developed. For the individual growth model, we simply use the implicit Runge-Kutta method. For the population dynamic model of partial differential problem, we use a characteristic finite difference method in the age-time domain and a finite element method with numerical integration and upwind modification in the spatial domain. Furthermore, the numerical scheme has been proved to yield numerical approximations with optimal error estimates and produce biologically reasonable approximate solutions as well. The mathematical and computational models have been used to study a specific model of a population of rainbow trout, Oncorhynchus mykiss, in a spatial environment. We Have investigated numerically the dynamics of spatio-temporal population distribution variations as they are viewed through the fish population density, total fish biomass, total fish age, total fish lipid, total fish structure (protein) and total fish protected protein. Furthermore, the model has also been used to study the effects of a spatially distributed nonpolar narcotic chemical on a rainbow trout population. The combined effects of lethal and sublethal toxicant effects have been considered. The methodologies and conclusions in this dissertation can be extended immediately into other populations and even some community settings, such as the fish-Daphnia predator-prey model if Daphnia are assumed to be immobile

Potential Theory on Lipschitz Domains in Riemannian Manifolds: Sobolev–Besov Space Results and the Poisson Problem

We continue a program to develop layer potential techniques for PDE on Lipschitz domains in Riemannian manifolds. Building on Lp and Hardy space estimates established in previous papers, here we establish Sobolev and Besov space estimates on solutions to the Dirichlet and Neumann problems for the Laplace operator plus a potential, on a Lipschitz domain in a Riemannian manifold with a metric tensor smooth of class C1+γ, for some γ>0. We treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant-coefficient Laplace operator on a Lipschitz domain in Euclidean space, with the Dirichlet boundary condition, by D. Jerison and C. Kenig

Research in the general area of non-linear dynamical systems Final report, 8 Jun. 1965 - 8 Jun. 1967

Nonlinear dynamical systems research on systems stability, invariance principles, Liapunov functions, and Volterra and functional integral equation