Inverse Inequality Estimates with Symbolic Computation

TL;DR

This paper improves eigenvalue bounds in PDE numerical analysis by developing a novel symbolic computation method for determinants, leading to tighter estimates and better understanding of asymptotic behavior.

Contribution

It introduces a modified holonomic ansatz applicable to a broader class of determinants, enabling explicit formulas and improved eigenvalue bounds in finite element analysis.

Findings

Improved upper bound on maximal eigenvalues by a factor of 8.

Derived explicit closed-form determinant expressions.

Established new tight bounds and asymptotic behavior for eigenvalues.

Abstract

In the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurately as possible. We apply symbolic computation methods to the situation of square elements and are able to improve the previously known upper bound, given in "p- and hp-finite element methods" (Schwab, 1998), by a factor of 8. More precisely, we try to evaluate the corresponding determinant using the holonomic ansatz, which is a powerful tool for dealing with determinants, proposed by Zeilberger in 2007. However, it turns out that this method does not succeed on the problem at hand. As a solution we present a variation of the original holonomic ansatz that is applicable to a larger class of determinants, including the one we are dealing with here. We obtain an explicit closed form for the determinant, whose special form enables us to derive new and tight upper resp. lower bounds on the maximal eigenvalue, as well as its asymptotic behaviour.