Numerical Methods For Eigenvalue Optimization Problems Of Some Elliptic And Biharmonie Operators

This dissertation studies the eigenvalue optimization problems of three linear dif-ferential operators. Firstly, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Then we discuss how to design the potential function with constraint to minimize and maxi-mize the least eigenvalue to the Schrodinger equation. At last, we deal with the extremal eigenvalue problem of the bi-harmonic operator with Navier boundary condition and mass constraint.In Chapter1, we give a brief introduction of the spectrum optimization problem. We review the existing theoretical results and numerical methods for the eigenvalue and shape optimization problems, then list the mathematical models we will study in this paper.In Chapter2, we simply recall the spectral theory to elliptic operators and the computational methods for the eigenvalue of the sparse matrix.In Chapter3, an extremal eigenvalue problem to the Sturm-Liouville equations with discontinuous coefficients and volume constraint is investigated. Liouville transforma-tion is applied to reformulate the problem into an equivalent minimization problem. Finite element method is proposed and the convergence for the finite element solution is established. A monotonie decreasing algorithm is presented to solve the extremal eigenvalue problem. A global convergence for the algorithm in the continuous case is proved. Numerical results are given to depict the efficiency of the method.In Chapter4, we discuss how to design the potential function with constraint to minimize or maximize the least eigenvalue to the Schrodinger operator. For the mini-mization case, we prove the extremal eigenvalue λ1(V,u) is equivalent to the fixed point about the potential function V and the eigenfunction u. According to this property, we propose a monotonie decreasing algorithm. And we extend this optimization method to the maximization case. We apply the finite element method in two optimization problems and numerical experiments show our algorithm is efficient.In Chapter5. we consider the extremal eigenvalue problem of the bi-haannonic operator with Xavier boundary condition and mass constraint.We use the Ciarlet-Raviart mixed finite element method to solve this equation,then employ monotone algorithm in this optimizat ion problem. Numerical examples demonstrate t he feasibility of the algorithm.Filially,we summarize the dissertation and propose the future research work.