University of Arizona

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# Analysis of the Tau Method for Eigenvalue Problems

Partial differential equations (PDEs) arise in numerous applications across science and engineering, and numerical simulations are often essential. Spectral methods are particularly effective for solving PDEs on simple geometries, providing highly accurate numerical solutions for a wide range of equations. Among these methods, the tau method, first introduced by C. Lanczos in 1938 for function approximation, is a valuable tool. The main idea of this technique is to approximate the solution of a given problem by solving exactly an approximate problem that incorporates a perturbation term, known as the "tau correction". Recent studies have highlighted the effectiveness of tau methods for various problems, while also revealing that the choice of the perturbation term significantly influences solution properties. However, there is currently no established criterion for selecting this term optimally.

In this work, we analyze two eigenvalue problems using a one-parameter family of methods known as the Gegenbauer tau methods, considering different formulations of the method. We demonstrate that with an appropriate parameter selection, it is possible to avoid spurious eigenvalues and achieve higher accuracy in eigenvalue computation, contributing to more reliable numerical solutions.