A new formulation for the numerical proof of the existence of solutions to elliptic problems

TL;DR

This paper introduces a novel operator matrix representation for infinite-dimensional Newton methods, improving the efficiency of numerical proofs of solutions to elliptic PDEs.

Contribution

It proposes a new operator matrix decomposition of the inverse linearized operator, enhancing verification procedures for elliptic PDE solutions.

Findings

More efficient verification compared to existing methods

Numerical examples confirm the method's usefulness

Operator matrix representation improves computational performance

Abstract

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {\mathcal L}^{-1} {\mathcal F}(\hat{u}) + {\mathcal L}^{-1} {\mathcal G}(w)$, where ${\mathcal L}$ is a linearized operator, ${\mathcal F}(\hat{u})$ is a residual, and ${\mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $\| {\mathcal L}^{-1} {\mathcal F}(\hat{u}) \|$ and $\| {\mathcal L}^{-1}{\mathcal G}(w) \|$ play major roles in the verification procedures. In this paper, using a similar concept as the `Schur complement' for matrix problems, we represent the inverse operator ${\mathcal L}^{-1}$ as an infinite-dimensional operator matrix that can be decomposed into two parts, one finite dimensional and one infinite dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, enabling a more efficient verification procedure compared with existing methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as ${\mathcal L}^{-1}$ are presented in the appendix.