High-Performance Computing for Spectral Approximations

Abstract

In this chapter, we focus on the numerical solution of large eigenvalue problems arising in finite-rank discretizations of integral operators.

Let X be a Banach space over ℂ and T a compact linear operator defined on X. We aim to solve numerically the eigenvalue problem

$$T\varphi = \lambda\varphi,$$

with λ nonzero and ϕ defined in X. Approximations λ _m and ϕ _m for the spectral elements of the integral operator can be obtained by solving

$$T_{m\varphi m} = \theta_{m\varphi m},$$

where (T _m ) is a sequence of finite-rank operators converging to T [AhLa01]. By evaluating the projected problem on a specific basis function, it is reduced to a matrix spectral problem

$$A_m x_m = \theta_m x_m$$

(33.1)

for a finite matrix A _m [AhLa06].