TY - JOUR

T1 - A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems

T2 - without direct estimation for an inverse of the linearized operator

AU - Sekine, Kouta

AU - Nakao, Mitsuhiro T.

AU - Oishi, Shin’ichi

N1 - Funding Information:
This work was supported by JST CREST Grant Number JPMJCR14D4 and MEXT under the “Exploratory Issue on Post-K computer” project (Development of verified numerical computations and super high-performance computing environment for extreme research). The second author was supported by JSPS KAKENHI Grant Number 18K03434. We thank the editors and reviewers for providing useful comments that helped to improve the content of this manuscript.

PY - 2020

Y1 - 2020

N2 - 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^) + L- 1G(w) , where L is a linearized operator, F(u^) is a residual, and G(w) is a nonlinear term. Therefore, the estimations of ‖ L- 1F(u^) ‖ and ‖ L- 1G(w) ‖ 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”.

AB - 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^) + L- 1G(w) , where L is a linearized operator, F(u^) is a residual, and G(w) is a nonlinear term. Therefore, the estimations of ‖ L- 1F(u^) ‖ and ‖ L- 1G(w) ‖ 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”.

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U2 - 10.1007/s00211-020-01155-7

DO - 10.1007/s00211-020-01155-7

M3 - Article

AN - SCOPUS:85093838172

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

ER -