Blog
About

1
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

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

      Preprint

      , ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          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.

          Related collections

          Most cited references 10

          • Record: found
          • Abstract: not found
          • Article: not found

          A numerical approach to the proof of existence of solutions for elliptic problems

            Bookmark
            • Record: found
            • Abstract: not found
            • Article: not found

            A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

              Bookmark
              • Record: found
              • Abstract: not found
              • Article: not found

              An Efficient Approach to the Numerical Verification for Solutions of Elliptic Differential Equations

                Bookmark

                Author and article information

                Journal
                01 October 2019
                Article
                1910.00759

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                65G20, 65N30, 35J25
                16 page, 1 figure
                math.NA cs.NA math.AP math.FA

                Analysis, Numerical & Computational mathematics, Functional analysis

                Comments

                Comment on this article