Efficient Computational Methods For The Helmholtz Transmission Eigenvalue Problem

The Helmholtz transmission eigenvalue problems arise in inverse scattering theory for an inhomogeneous medium.They have widely physical background,for example,they can be used to obtain estimates for the material properties of the scattering object and have theoretical importance in the uniqueness and reconstruction in inverse scattering theory.The transmission eigenvalue problems are nonlinear quadratic eigenvalue problems so that the research on their computational methods and theoretical analysis are much more difficult than the linear eigenvalue problems.Due to their importance,The transmission eigenvalue problems attract many scholars' attention in the relevant fields.Their research interest mainly centers on the theoretical issues such as the existence of transmission eigenvalues and the upper and lower bounds for the index of refraction.In recent years,the numerical methods of the transmission eigenvalue problem are hot topics in the field of engineering and computational mathematics.The first numerical study was made by Colton et al.in 2010 and involves three numerical methods including Argynis method,continuous method,divergence-free method.Later on,their work is attached great importance to in computational mathematics,and many efficient methods are developed,such as iterative method,multigrid method,spectral method and mixed method.Among them,the iterative method needs an iterative initial value which is an approximation of the transmission eigenvalue.This method is valid for the real eigenvalues in practical computation and its convergence is verified theoretically under the assumption on the simple eigenvalue.The multigrid method in existing literature is based on the iterative method and so the similar issues do happen.The spectral method is efficient for problems on separable domains such as circle,rectangle and circular cylinder but lacks domain flexibility and circular cylinder.The existing mixed method is highly efficient but lack strict theoretical analysis.This dissertation is devoted to the efficient computational methods for the Helmholtz transmission eigenvalue problem.The main contents of this paper include the following.In Chapter 1,we introduce the preliminaries of finite element method of the Helmholtz transmission eigenvalue problem.We review the classical weak form of the Helmholtz transmission eigenvalue problem.We linearize it into an equivalent nonsymmetric eigenvalue problem.The linear weak form is the foundation for the efficient numerical algorithms.Based on the linear weak form,we further introduce an H2-conforming element discretization,which is algebraically equivalent to the Argyris element discretization proposed by Colton et al.In Chapter 2,based on conforming element discretization,we use multilevel correction technique to establish a new multigrid scheme.With this scheme,the solution of eigenvalue problem on a fine mesh can be reduced to a series of the solutions of the eigenvalue problem on a coarse mesh and a series of solutions of the boundary value problems on the multilevel meshes.Finally we give the theoretical analysis and some numerical examples.In Chapter 3,in the aid of the theory of a posteriori estimates on fourth order problem established by Verf¨urth,we research the a posteriori estimates and adaptive algorithm for the transmission eigenvalue problem.This is a new work that involves the main stream in efficient computation.We give the a posteriori error estimators for primal and dual eigenfunctions,and prove their effectiveness and reliability.According to the basic relation between eigenvalue and approximate eigenvalue,we shall give the a posteriori error estimator for eigenvalues and design an adaptive algorithm to solve for complex and multiple eigenvalues.In numerical experiments,we adopt Argyris element to realize our algorithm.In the completed numerical examples,we adopt the Argyris element to implement the adaptive algorithm and numerical results not only confirm the efficiency and reliability of the indicator but also indicate that in many cases the error of numerical eigenvalues can achieve the optimal convergence order even on a nonconvex domain.In Chapter 4,we propose an Hm-conforming spectral element method on multidimensional domain(m ? 1)and apply it to compute the transmission eigenvalues.We organize our research as follows:(1)Consider the construction of bases on the standard interval [-1,1].According to the property of generalized Jacobi polynomial,we can use Legendre or Chebyshev polynomial to construct the buble functions on [-1,1],which together with their low order derivatives vanish at-1 and 1.To guarantee the Hm-conformity of spectral elements,the nodal bases associated to the nodal function value and low order derivative value are constructed.In addition,the spectral element interpolation operator is constructed to prove the interpolation error estimates.(2)The tensor products of one dimensional bases and interpolation operators are respectively adopted to be the bases and interpolation operator on the standard element on multi-dimensional domain.The interpolation error estimates on the element on mulgtidimensional domain are proved using the ones on one dimension.The affine transform from standard element to general element are easily constructed.Hence the scaled argument can be used to prove the interpolation error estimates on general rectangle element.(3)The H2-conforming element method is applied to the transmission eigenvalue problem,and the error estimates for eigenpairs are proved.Some numerical experiments confirm the high efficiency of the spectral element method.In Chapter 5,as the supplement of the existing nonconforming element method of class H1,the nonconforming element method of class L2 is proposed,which can cover the discontinuous finite element spaces.We apply Babuska-Osborn spectral approximation theory in the union of the piecewise smooth Sobolev spaces to prove the optimal convergence order of numerical eigenvalues by this method.The numerical results show that the discretization of this paper using the cubic tetrahedron element mixed with linear element can capture the eigenvalues of higher accuracy than the existing discretization using the Morley-Zienkiewicz element.In Chapter 6,we introduce three other numerical algorithms and the associated numerical results under the existing finite element discretizations,including two grid discretization algorithm,nonconforming element discretization algorithm of class H1 and Ciarlet-Raviart spectral mixed discretiation algorithm.Numerical experiments are carried out and validate that the numerical eigenvalues obtained by the two formers can achieve the asymptotically optimal convergence rate and those obtained by the last one can provide superior accuracy.