Validated matrix multiplication transform for orthogonal polynomials with applications to computer-assisted proofs for PDEs

TL;DR

This paper introduces a validated matrix multiplication transform for orthogonal polynomials, enabling accurate spectral computations and computer-assisted proofs for nonlinear PDEs on complex geometries like disks.

Contribution

It presents a novel validated matrix multiplication transform algorithm and applies it to Zernike polynomials for rigorous PDE analysis on disks.

Findings

Validated MMT reduces rounding errors in spectral nonlinear computations

Efficient computation of Zernike polynomial products demonstrated

Computer-assisted proofs of PDE solutions established using MMT and Zernike series

Abstract

In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform (MMT) algorithm, analogous to the discrete Fourier transform, which offers a reliable framework for evaluating nonlinearities in spectral methods while effectively mitigating challenges associated with rounding errors. Second, we examine the Zernike polynomials, a spectral basis well-suited for problems on the disk, and highlight their essential properties. We further demonstrate how the MMT approach can be effectively employed to compute the product of truncated Zernike series, ensuring both accuracy and efficiency. Finally, we combine the MMT framework and Zernike series to construct computer-assisted proofs that establish the existence of solutions to two distinct nonlinear elliptic PDEs on the disk.