Richardson Extrapolation Of Eigenvalue Problems For Integral Equations

For the eigenvalue problem of integral equations of the second-kind ∫_Ωk(t,s)u(s)ds = λu(t), t,s ∈ Ω (?) Rⁿ, reference [1] established the asymptotic expansion of the approximated eigenvalue λ_h, for Problem (1) by means of iterative Galerkin finite element methods in certain piecewise polynomial spaces. Suppose that the Galerkin eigen-pair (λ_h,,u_h) of order m — 1 approximates (λ, u), [1] derived the asymptotic expansion of λ_h:where β_u depends only on u_h. Using the Richardson extrapolation for λ_h, the author obtained a higher order accuracy approximation of the simple eigenvalue λ:However, for semi-simple eigenvalues, such as eigenvalues of boundary integral equations and high dimensional integral equations in eigenvalue problems, u approximated by u_h and u approximated by u_h/2 are not necessarily the same. Therefore, we can't use themethod in [1] to get the higher order approximation of the semi-simple eigenvalue A directly. In this thesis, we derive a new formula to get the asymptotic expansion of the Galerkin method in case of piecewise finite element spaces. Let u_h^s be the Sloan iteration of u_h, i.e., λ_hu_h^s = Tu_h. Let , and . The first main result of this thesis is the following (Theorem 1):To have asymptotic expansions of λ — λ_h and λ — λ^s, we need...