Least-squares spectral element preconditioners for fourth order elliptic problems

TL;DR

This paper introduces spectral element preconditioners for fourth order elliptic problems that are easy to invert and demonstrate spectral equivalence, improving solution efficiency.

Contribution

It extends separation of variables-based preconditioners to fourth order elliptic problems, providing a new approach with proven spectral equivalence and validated numerical results.

Findings

Preconditioners are spectrally equivalent to the quadratic forms.

Numerical results validate theoretical estimates.

Preconditioners outperform existing methods for similar problems.

Abstract

The goal of this paper is to propose preconditioners for the system of linear equations that arises from a discretization of fourth order elliptic problems using spectral element methods. These preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert. For second order elliptic problems this technique has proven to be very successful and performs better than other preconditioners. We show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them. Numerical result for the biharmonic problem are presented to validate the theoretical estimates.