  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.
