A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: withou
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Numerische Mathematik
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 Kouta Sekine1 · Mitsuhiro T. Nakao2 · Shin’ichi Oishi3 Received: 31 May 2019 / Revised: 4 August 2020 / Accepted: 13 October 2020 © The Author(s) 2020
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 infiniteˆ + L−1 G(w), where dimensional Newton-type fixed point equation w = −L−1 F(u) L is a linearized operator, F(u) ˆ is a residual, and G(w) is a nonlinear term. Therefore, ˆ and L−1 G(w) play major roles in the verification the estimations of L−1 F(u) 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”. Mathematics Subject Classification 65G20 · 65N30 · 35J25
B
Kouta Sekine [email protected]
1
Faculty of Information Networking for Innovation and Design, Toyo University, 1-7-11 Akabanedai, kita-ku, Tokyo 115-0053, Japan
2
Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Tokyo 169-8555, Japan
3
Department of Applied Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Tokyo 169-8555, Japan
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K. Sekine et al.
1 Introduction In this paper, we study a new approach to proving the existence of solutions to elliptic problems. The proposed approach offers an improvement over existing Nakao’s methods that use a finite-dimensional projection, that is, FN and IN methods in [11] and [9, Part I]. Particularly, an important aspect of our approach is the formulation using the Schur complement without direct estimates of an inverse of the linearized operator. Our approach inherits the advantages of both Nakao’s FN and IN methods using a finite-dimensional projection, which also indicates that the disadvantages of both methods are resolved. We consider computer-assisted existence proofs for the nonlinear elliptic boundary value problem
−Δu = f (u) u=0
in Ω, on ∂Ω,
(1)
where Ω ⊂ Rn (n = 1, 2, 3) is a bounded domain with a Lipschitz boundary, and f : H01 (Ω) → H −1 (Ω) is a given nonlinear function which is assumed to be Fréchet differentiable. Equation (1) is a basic case of a semi-linear
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