Paper Info
Title
A New Formulation Using The Schur Complement For The Numerical Existence Proof Of Solutions To Elliptic Problems: Without Direct Estimation For An Inverse Of The Linearized Operator
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-dimensional Newton-type fixed point equation w = -L-1F((u) over cap) + L(-1)G(w), where L is a linearized operator, F((u) over cap) is a residual, and G(w) is a nonlinear term. Therefore, the estimations of parallel to L-1F((u) over cap)parallel to and parallel to L(-1)G(w)parallel to play major roles in the verification procedures. In this paper, using a similar concept to block Gaussian elimination and its corresponding 'Schur complement' for matrix problems, we represent the inverse operator L-1 as an infinite-dimensional operator matrix that can be decomposed into two parts: finite-dimensional and infinite-dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, which enables a more efficient verification procedure compared with existing Nakao's 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 L-1 are presented in the "Appendix".
Year
DOI
Venue
2020
10.1007/s00211-020-01155-7
NUMERISCHE MATHEMATIK
DocType
Volume
Issue
Journal
146
4
ISSN
Citations 
PageRank 
0029-599X
0
0.34
References 
Authors
0
3
Name
Order
Citations
PageRank
Kouta Sekine101.69
m nakao211.37
Shin'ichi Oishi328037.14