Lehmann–Goerisch Method for High-Precision Eigenvalue Bounds

This chapter provides a deep exploration of the Lehmann–Goerisch method, designed for the efficient computation of high-precision eigenvalue bounds. As a feature of this book, it discloses the affinity of the Lehmann–Goerisch method with finite element methods, which has not been well discussed in the existing literature. The implementation of the Lehmann–Goerisch method necessitates a rough lower bound for a specific eigenvalue, which can be obtained from the lower bound derived in Chaps. 3 and 4 . This method takes the advantage of accurate approximate eigenfunctions computed by any means and subsequently delivers precise eigenvalue bounds. The effectiveness of this method is demonstrated via its application to the Laplacian and Steklov eigenvalue problems.