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

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.